answer：Okay, so I have this problem: "In how many ways can 5 identical coins be placed in 3 different pockets?" Hmm, let's see. I've heard about combinatorics problems like this before, where you have identical items and distinct containers. I think it's something to do with combinations or permutations, but I'm not entirely sure which one applies here.First, let me try to understand the problem better. We have 5 coins, and they're all identical, meaning one coin is the same as another. So, it doesn't matter which specific coin goes into which pocket; what matters is how many coins end up in each pocket. The pockets are different, so putting 2 coins in the first pocket and 3 in the second is different from putting 3 in the first and 2 in the second.I remember something about the "stars and bars" theorem in combinatorics, which deals with distributing identical objects into distinct bins. Maybe that's what I need to use here. Let me recall how that works.The stars and bars theorem says that the number of ways to distribute 'n' identical objects into 'k' distinct bins is equal to the combination of (n + k - 1) choose (k - 1). So, in formula terms, it would be C(n + k - 1, k - 1), where C is the combination function.In this problem, n is 5 (the coins) and k is 3 (the pockets). So, plugging into the formula, it should be C(5 + 3 - 1, 3 - 1) = C(7, 2). Now, I need to calculate C(7, 2).C(7, 2) is calculated as 7! / (2! * (7 - 2)!) = (7 * 6 * 5!) / (2 * 1 * 5!) = (7 * 6) / 2 = 42 / 2 = 21.So, according to this, there are 21 ways to distribute 5 identical coins into 3 different pockets.But wait, let me make sure I'm not missing something. Is there any restriction on the number of coins per pocket? The problem doesn't specify any, so I assume that a pocket can have zero or more coins. If there were restrictions, like each pocket must have at least one coin, the formula would be slightly different. In that case, it would be C(n - 1, k - 1). But since there's no such restriction here, the original formula applies.Let me try to think of a simpler case to verify. Suppose I have 2 coins and 2 pockets. According to the formula, it should be C(2 + 2 - 1, 2 - 1) = C(3, 1) = 3 ways. Let's list them:1. Both coins in the first pocket.2. Both coins in the second pocket.3. One coin in each pocket.Yes, that's 3 ways, which matches the formula. So, it seems to work for this simpler case.Another example: 3 coins and 2 pockets. The formula gives C(3 + 2 - 1, 2 - 1) = C(4, 1) = 4 ways. Listing them:1. All 3 coins in the first pocket.2. All 3 coins in the second pocket.3. 2 coins in the first, 1 in the second.4. 1 coin in the first, 2 in the second.Again, that's 4 ways, which matches the formula. So, it seems consistent.Therefore, going back to the original problem, with 5 coins and 3 pockets, the number of ways should indeed be 21.I think I've got it. The key was recognizing that this is a stars and bars problem where we're distributing identical objects into distinct bins without any restrictions on the number per bin. Applying the formula gives the answer.

answer：First, I notice that both terms on the left side of the equation have a common factor of ( x! ). I'll factor that out to simplify the equation.After factoring, the equation becomes ( (x + 2) cdot x! = 40320 ). I recognize that ( 40320 ) is equal to ( 8! ), so I can rewrite the equation as ( (x + 2)! = 8! ).Since the factorials are equal, their arguments must be equal as well. Therefore, ( x + 2 = 8 ). Solving for ( x ) gives ( x = 6 ).

answer：Okay, so I have this geometry problem here, and I need to figure out the area of quadrilateral BEFC in triangle ABC. Let me try to visualize and break it down step by step.First, the problem says that ABC is an isosceles triangle with AB = AC = 3 cm and BC = 4 cm. So, ABC has two equal sides, AB and AC, each 3 cm, and the base BC is 4 cm. I can sketch this triangle with A at the top, B and C at the base, and the base BC extended to point D such that CD equals BC, which is 4 cm. So, BD would be BC + CD = 4 + 4 = 8 cm.Next, E is the midpoint of AB. Since AB is 3 cm, the midpoint E would divide AB into two equal parts, each 1.5 cm. So, AE = EB = 1.5 cm.There's a circle centered at E with radius EB, which is 1.5 cm. This circle passes through B and intersects AC at point F above C. So, F is somewhere on AC closer to C because the circle is centered at E, which is closer to B.I need to find the area of quadrilateral BEFC. To do this, maybe I can find the coordinates of all the points involved and then use coordinate geometry to calculate the area. Let me assign coordinates to the triangle ABC.Let me place point B at (-2, 0) and point C at (2, 0) so that BC is 4 cm. Since ABC is isosceles with AB = AC, point A must lie somewhere along the y-axis. Let me find the coordinates of A.Using the distance formula, the length of AB is 3 cm. So, if A is at (0, h), then the distance from A to B is sqrt[(0 - (-2))² + (h - 0)²] = sqrt[4 + h²] = 3. So, 4 + h² = 9 => h² = 5 => h = sqrt(5). Therefore, point A is at (0, sqrt(5)).So, coordinates:- A: (0, sqrt(5))- B: (-2, 0)- C: (2, 0)Now, E is the midpoint of AB. The midpoint formula gives E as ((-2 + 0)/2, (0 + sqrt(5))/2) = (-1, sqrt(5)/2).The circle centered at E with radius EB. Let's compute EB. Since E is (-1, sqrt(5)/2) and B is (-2, 0), the distance EB is sqrt[(-1 - (-2))² + (sqrt(5)/2 - 0)²] = sqrt[(1)² + (sqrt(5)/2)²] = sqrt[1 + 5/4] = sqrt[9/4] = 3/2. So, the radius of the circle is 3/2 cm.The equation of the circle centered at E is (x + 1)² + (y - sqrt(5)/2)² = (3/2)² = 9/4.Now, I need to find the point F where this circle intersects AC. Let me find the equation of AC. Points A (0, sqrt(5)) and C (2, 0). The slope of AC is (0 - sqrt(5))/(2 - 0) = -sqrt(5)/2. So, the equation is y - sqrt(5) = (-sqrt(5)/2)(x - 0), which simplifies to y = (-sqrt(5)/2)x + sqrt(5).So, the equation of AC is y = (-sqrt(5)/2)x + sqrt(5).Now, to find point F, solve the system of equations:1. (x + 1)² + (y - sqrt(5)/2)² = 9/42. y = (-sqrt(5)/2)x + sqrt(5)Substitute equation 2 into equation 1:(x + 1)² + [(-sqrt(5)/2 x + sqrt(5) - sqrt(5)/2)]² = 9/4Simplify the y-component:(-sqrt(5)/2 x + sqrt(5) - sqrt(5)/2) = (-sqrt(5)/2 x + sqrt(5)/2)So, the equation becomes:(x + 1)² + [(-sqrt(5)/2 x + sqrt(5)/2)]² = 9/4Let me compute each part:First, (x + 1)² = x² + 2x + 1Second, [(-sqrt(5)/2 x + sqrt(5)/2)]²:= [(-sqrt(5)/2 x + sqrt(5)/2)]²= (sqrt(5)/2)^2 ( -x + 1 )²= (5/4)(x² - 2x + 1)So, putting it all together:x² + 2x + 1 + (5/4)(x² - 2x + 1) = 9/4Multiply both sides by 4 to eliminate denominators:4x² + 8x + 4 + 5(x² - 2x + 1) = 9Expand:4x² + 8x + 4 + 5x² - 10x + 5 = 9Combine like terms:(4x² + 5x²) + (8x - 10x) + (4 + 5) = 99x² - 2x + 9 = 9Subtract 9 from both sides:9x² - 2x = 0Factor:x(9x - 2) = 0So, x = 0 or x = 2/9.Now, x = 0 would correspond to point A, but we need the other intersection point, which is F. So, x = 2/9.Now, find y-coordinate of F using equation of AC:y = (-sqrt(5)/2)(2/9) + sqrt(5)= (-sqrt(5)/9) + sqrt(5)= ( -sqrt(5) + 9 sqrt(5) ) / 9= (8 sqrt(5))/9So, point F is at (2/9, 8 sqrt(5)/9).Now, I have coordinates for points B, E, F, and C. Let me list them:- B: (-2, 0)- E: (-1, sqrt(5)/2)- F: (2/9, 8 sqrt(5)/9)- C: (2, 0)To find the area of quadrilateral BEFC, I can use the shoelace formula. Let me order the points in a cyclic order. Let's go from B to E to F to C and back to B.So, the coordinates in order are:1. B: (-2, 0)2. E: (-1, sqrt(5)/2)3. F: (2/9, 8 sqrt(5)/9)4. C: (2, 0)Let me apply the shoelace formula:Area = 1/2 |sum over i (x_i y_{i+1} - x_{i+1} y_i)|Compute each term:Term 1: x_B y_E - x_E y_B = (-2)(sqrt(5)/2) - (-1)(0) = -sqrt(5) - 0 = -sqrt(5)Term 2: x_E y_F - x_F y_E = (-1)(8 sqrt(5)/9) - (2/9)(sqrt(5)/2) = (-8 sqrt(5)/9) - (sqrt(5)/9) = (-9 sqrt(5)/9) = -sqrt(5)Term 3: x_F y_C - x_C y_F = (2/9)(0) - (2)(8 sqrt(5)/9) = 0 - 16 sqrt(5)/9 = -16 sqrt(5)/9Term 4: x_C y_B - x_B y_C = (2)(0) - (-2)(0) = 0 - 0 = 0Sum of all terms: (-sqrt(5)) + (-sqrt(5)) + (-16 sqrt(5)/9) + 0 = (-2 sqrt(5)) - (16 sqrt(5)/9)Convert to common denominator:-2 sqrt(5) = -18 sqrt(5)/9So, total sum: (-18 sqrt(5)/9 - 16 sqrt(5)/9) = (-34 sqrt(5)/9)Take absolute value: 34 sqrt(5)/9Multiply by 1/2: (34 sqrt(5)/9) * (1/2) = 17 sqrt(5)/9So, the area of quadrilateral BEFC is 17 sqrt(5)/9 cm².Wait, let me double-check my calculations because I might have made a mistake in the shoelace formula.Let me list the coordinates again:B: (-2, 0)E: (-1, sqrt(5)/2)F: (2/9, 8 sqrt(5)/9)C: (2, 0)Shoelace formula steps:Compute sum of x_i y_{i+1}:1. x_B y_E = (-2)(sqrt(5)/2) = -sqrt(5)2. x_E y_F = (-1)(8 sqrt(5)/9) = -8 sqrt(5)/93. x_F y_C = (2/9)(0) = 04. x_C y_B = (2)(0) = 0Total sum: -sqrt(5) -8 sqrt(5)/9 + 0 + 0 = (-9 sqrt(5)/9 -8 sqrt(5)/9) = -17 sqrt(5)/9Compute sum of y_i x_{i+1}:1. y_B x_E = (0)(-1) = 02. y_E x_F = (sqrt(5)/2)(2/9) = (sqrt(5)/2)(2/9) = sqrt(5)/93. y_F x_C = (8 sqrt(5)/9)(2) = 16 sqrt(5)/94. y_C x_B = (0)(-2) = 0Total sum: 0 + sqrt(5)/9 + 16 sqrt(5)/9 + 0 = 17 sqrt(5)/9Now, subtract the two sums:|(-17 sqrt(5)/9 - 17 sqrt(5)/9)| = |-34 sqrt(5)/9| = 34 sqrt(5)/9Multiply by 1/2: 17 sqrt(5)/9Yes, that's correct. So, the area is 17 sqrt(5)/9 cm².But wait, let me think again. The shoelace formula should be 1/2 |sum(x_i y_{i+1} - x_{i+1} y_i)|. I think I might have messed up the order or the signs.Wait, in my first calculation, I think I did it correctly, but let me re-express it properly.List the coordinates in order: B, E, F, C, B.Compute x_i y_{i+1}:B to E: (-2)(sqrt(5)/2) = -sqrt(5)E to F: (-1)(8 sqrt(5)/9) = -8 sqrt(5)/9F to C: (2/9)(0) = 0C to B: (2)(0) = 0Total: -sqrt(5) -8 sqrt(5)/9Compute y_i x_{i+1}:B to E: (0)(-1) = 0E to F: (sqrt(5)/2)(2/9) = sqrt(5)/9F to C: (8 sqrt(5)/9)(2) = 16 sqrt(5)/9C to B: (0)(-2) = 0Total: sqrt(5)/9 + 16 sqrt(5)/9 = 17 sqrt(5)/9Now, subtract the two sums:(-sqrt(5) -8 sqrt(5)/9) - (17 sqrt(5)/9) = (-9 sqrt(5)/9 -8 sqrt(5)/9 -17 sqrt(5)/9) = (-34 sqrt(5)/9)Take absolute value: 34 sqrt(5)/9Multiply by 1/2: 17 sqrt(5)/9Yes, that's consistent. So, the area is 17 sqrt(5)/9 cm².But let me check if this makes sense. The area of triangle ABC is 2 sqrt(5), which is approximately 4.472 cm². The quadrilateral BEFC is a part of this triangle, so 17 sqrt(5)/9 is approximately (17*2.236)/9 ≈ 38.012/9 ≈ 4.223 cm², which is almost the entire area of ABC. That seems too large because BEFC is a quadrilateral inside ABC, so it should be less than the area of ABC.Wait, that can't be right. Maybe I made a mistake in the shoelace formula.Wait, let me recast the coordinates:B: (-2, 0)E: (-1, sqrt(5)/2)F: (2/9, 8 sqrt(5)/9)C: (2, 0)Let me plot these points approximately:- B is at (-2, 0)- E is at (-1, approx 1.118)- F is at (approx 0.222, approx 3.97)- C is at (2, 0)Wait, point F is at (2/9, 8 sqrt(5)/9). Let's compute 8 sqrt(5)/9:sqrt(5) ≈ 2.236, so 8*2.236 ≈ 17.888, divided by 9 ≈ 1.987. So, F is at approximately (0.222, 1.987).Wait, that makes more sense. So, F is somewhere on AC closer to C.So, the quadrilateral BEFC is a four-sided figure with vertices at B (-2,0), E (-1, ~1.118), F (~0.222, ~1.987), and C (2,0). So, it's a quadrilateral that covers a significant portion of the triangle ABC but not the entire area.Wait, the area I calculated was 17 sqrt(5)/9 ≈ 4.223 cm², but the area of ABC is 2 sqrt(5) ≈ 4.472 cm². So, BEFC is almost the entire triangle except a small part near A. That seems plausible because F is close to C, so BEFC covers most of the triangle.But let me think if there's another way to compute the area, maybe by subtracting areas.Alternatively, I can compute the area of triangle BEF and the area of triangle EFC and add them together.First, find the area of triangle BEF.Points B (-2,0), E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9).Using the shoelace formula for triangle BEF:Coordinates:B: (-2, 0)E: (-1, sqrt(5)/2)F: (2/9, 8 sqrt(5)/9)Back to B: (-2, 0)Compute sum of x_i y_{i+1}:-2*(sqrt(5)/2) + (-1)*(8 sqrt(5)/9) + (2/9)*0 = -sqrt(5) -8 sqrt(5)/9 + 0 = (-9 sqrt(5)/9 -8 sqrt(5)/9) = -17 sqrt(5)/9Sum of y_i x_{i+1}:0*(-1) + (sqrt(5)/2)*(2/9) + (8 sqrt(5)/9)*(-2) = 0 + sqrt(5)/9 -16 sqrt(5)/9 = (-15 sqrt(5)/9) = -5 sqrt(5)/3Subtract the two sums:(-17 sqrt(5)/9) - (-5 sqrt(5)/3) = (-17 sqrt(5)/9 + 15 sqrt(5)/9) = (-2 sqrt(5)/9)Take absolute value and multiply by 1/2:| -2 sqrt(5)/9 | * 1/2 = (2 sqrt(5)/9) * 1/2 = sqrt(5)/9So, area of triangle BEF is sqrt(5)/9 cm².Now, find the area of triangle EFC.Points E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2, 0).Using shoelace formula:Coordinates:E: (-1, sqrt(5)/2)F: (2/9, 8 sqrt(5)/9)C: (2, 0)Back to E: (-1, sqrt(5)/2)Compute sum of x_i y_{i+1}:-1*(8 sqrt(5)/9) + (2/9)*0 + 2*(sqrt(5)/2) = -8 sqrt(5)/9 + 0 + sqrt(5) = (-8 sqrt(5)/9 + 9 sqrt(5)/9) = sqrt(5)/9Sum of y_i x_{i+1}:(sqrt(5)/2)*(2/9) + (8 sqrt(5)/9)*2 + 0*(-1) = (sqrt(5)/9) + 16 sqrt(5)/9 + 0 = 17 sqrt(5)/9Subtract the two sums:sqrt(5)/9 - 17 sqrt(5)/9 = (-16 sqrt(5)/9)Take absolute value and multiply by 1/2:| -16 sqrt(5)/9 | * 1/2 = (16 sqrt(5)/9) * 1/2 = 8 sqrt(5)/9So, area of triangle EFC is 8 sqrt(5)/9 cm².Now, total area of quadrilateral BEFC is area of BEF + area of EFC = sqrt(5)/9 + 8 sqrt(5)/9 = 9 sqrt(5)/9 = sqrt(5) cm².Wait, that's different from the shoelace formula result earlier. Which one is correct?Wait, when I used the shoelace formula on quadrilateral BEFC, I got 17 sqrt(5)/9 ≈ 4.223 cm², but when I split it into two triangles, I got sqrt(5) ≈ 2.236 cm². These are conflicting results. There must be a mistake in one of the methods.Let me check the shoelace formula again for quadrilateral BEFC.Coordinates in order: B (-2,0), E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2,0)Compute sum of x_i y_{i+1}:B to E: (-2)(sqrt(5)/2) = -sqrt(5)E to F: (-1)(8 sqrt(5)/9) = -8 sqrt(5)/9F to C: (2/9)(0) = 0C to B: (2)(0) = 0Total: -sqrt(5) -8 sqrt(5)/9Sum of y_i x_{i+1}:B to E: (0)(-1) = 0E to F: (sqrt(5)/2)(2/9) = sqrt(5)/9F to C: (8 sqrt(5)/9)(2) = 16 sqrt(5)/9C to B: (0)(-2) = 0Total: sqrt(5)/9 + 16 sqrt(5)/9 = 17 sqrt(5)/9Now, subtract the two sums:(-sqrt(5) -8 sqrt(5)/9) - (17 sqrt(5)/9) = (-9 sqrt(5)/9 -8 sqrt(5)/9 -17 sqrt(5)/9) = (-34 sqrt(5)/9)Take absolute value and multiply by 1/2:| -34 sqrt(5)/9 | * 1/2 = (34 sqrt(5)/9) * 1/2 = 17 sqrt(5)/9 ≈ 4.223 cm²But when I split into two triangles, I got sqrt(5) ≈ 2.236 cm². Clearly, one of these is wrong.Wait, perhaps I made a mistake in the shoelace formula for the quadrilateral. Let me check the order of the points again.In the shoelace formula, the order of the points must be either clockwise or counterclockwise without crossing. Let me ensure that the points are ordered correctly.Quadrilateral BEFC: B (-2,0), E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2,0). Plotting these, the order is correct in a counterclockwise manner.Alternatively, maybe I should have ordered them differently. Let me try another order: B, F, E, C.Wait, no, that might cross over. Let me stick to the original order.Alternatively, perhaps I made a mistake in the area of triangle EFC. Let me recalculate that.Area of triangle EFC using coordinates E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2, 0).Using shoelace formula:List the points: E, F, C, E.Compute sum of x_i y_{i+1}:E to F: (-1)(8 sqrt(5)/9) = -8 sqrt(5)/9F to C: (2/9)(0) = 0C to E: (2)(sqrt(5)/2) = sqrt(5)Total: -8 sqrt(5)/9 + 0 + sqrt(5) = (-8 sqrt(5)/9 + 9 sqrt(5)/9) = sqrt(5)/9Sum of y_i x_{i+1}:E to F: (sqrt(5)/2)(2/9) = sqrt(5)/9F to C: (8 sqrt(5)/9)(2) = 16 sqrt(5)/9C to E: (0)(-1) = 0Total: sqrt(5)/9 + 16 sqrt(5)/9 + 0 = 17 sqrt(5)/9Subtract the two sums:sqrt(5)/9 - 17 sqrt(5)/9 = -16 sqrt(5)/9Take absolute value and multiply by 1/2:| -16 sqrt(5)/9 | * 1/2 = (16 sqrt(5)/9) * 1/2 = 8 sqrt(5)/9So, area of EFC is 8 sqrt(5)/9 cm².Similarly, area of BEF was sqrt(5)/9 cm².Adding them together: 8 sqrt(5)/9 + sqrt(5)/9 = 9 sqrt(5)/9 = sqrt(5) cm².But this contradicts the shoelace formula result for the quadrilateral. So, which one is correct?Wait, perhaps the issue is that when I used the shoelace formula for the quadrilateral, I included the area outside the triangle ABC, but since F is inside ABC, the shoelace formula should give the correct area.Wait, let me compute the area of ABC using shoelace formula to confirm.Coordinates of ABC: A (0, sqrt(5)), B (-2,0), C (2,0), back to A.Compute sum of x_i y_{i+1}:0*0 + (-2)*0 + 2*sqrt(5) = 0 + 0 + 2 sqrt(5) = 2 sqrt(5)Sum of y_i x_{i+1}:sqrt(5)*(-2) + 0*2 + 0*0 = -2 sqrt(5) + 0 + 0 = -2 sqrt(5)Subtract the two sums:2 sqrt(5) - (-2 sqrt(5)) = 4 sqrt(5)Take absolute value and multiply by 1/2:|4 sqrt(5)| * 1/2 = 2 sqrt(5) cm², which matches the given area.Now, if the area of BEFC is sqrt(5) cm², which is approximately 2.236 cm², and the area of ABC is 2 sqrt(5) ≈ 4.472 cm², then the remaining area would be approximately 2.236 cm², which is plausible.But according to the shoelace formula for quadrilateral BEFC, I got 17 sqrt(5)/9 ≈ 4.223 cm², which is almost the entire area of ABC. That can't be right because F is inside ABC, so BEFC should be less than ABC.Wait, perhaps I made a mistake in the order of the points for the shoelace formula. Let me try a different order.Let me order the points as B, E, F, C, but ensure that the polygon doesn't intersect itself.Alternatively, maybe I should have ordered them as B, E, F, C, but in a way that the polygon is convex.Wait, let me try another approach. Instead of using shoelace formula, maybe I can use vectors or coordinate geometry to find the area.Alternatively, I can find the area of quadrilateral BEFC by subtracting the area of triangle AFE from the area of triangle ABC.Wait, but I don't know the area of AFE yet.Alternatively, since E is the midpoint of AB, and the circle centered at E with radius EB intersects AC at F, maybe there's a way to find the ratio AF/FC.Wait, in the circle, since E is the center and EB = EF = 3/2 cm, triangle EBF is an equilateral triangle? Wait, no, because EB = EF, but angle at E is not necessarily 60 degrees.Wait, maybe I can use coordinates again.Wait, I think the issue is that when I used the shoelace formula for the quadrilateral, I might have included an area outside the triangle, but since all points are inside ABC, that shouldn't be the case.Wait, let me recast the coordinates and try the shoelace formula again carefully.Points in order: B (-2,0), E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2,0)Compute x_i y_{i+1}:B to E: (-2)(sqrt(5)/2) = -sqrt(5)E to F: (-1)(8 sqrt(5)/9) = -8 sqrt(5)/9F to C: (2/9)(0) = 0C to B: (2)(0) = 0Total: -sqrt(5) -8 sqrt(5)/9Compute y_i x_{i+1}:B to E: (0)(-1) = 0E to F: (sqrt(5)/2)(2/9) = sqrt(5)/9F to C: (8 sqrt(5)/9)(2) = 16 sqrt(5)/9C to B: (0)(-2) = 0Total: sqrt(5)/9 + 16 sqrt(5)/9 = 17 sqrt(5)/9Now, subtract the two sums:(-sqrt(5) -8 sqrt(5)/9) - (17 sqrt(5)/9) = (-9 sqrt(5)/9 -8 sqrt(5)/9 -17 sqrt(5)/9) = (-34 sqrt(5)/9)Take absolute value and multiply by 1/2:| -34 sqrt(5)/9 | * 1/2 = (34 sqrt(5)/9) * 1/2 = 17 sqrt(5)/9 ≈ 4.223 cm²But as I thought earlier, this can't be correct because it's almost the entire area of ABC. Therefore, I must have made a mistake in the shoelace formula.Wait, perhaps I should have ordered the points differently. Let me try ordering them as B, C, F, E.So, points: B (-2,0), C (2,0), F (2/9, 8 sqrt(5)/9), E (-1, sqrt(5)/2), back to B.Compute x_i y_{i+1}:B to C: (-2)(0) = 0C to F: (2)(8 sqrt(5)/9) = 16 sqrt(5)/9F to E: (2/9)(sqrt(5)/2) = sqrt(5)/9E to B: (-1)(0) = 0Total: 0 + 16 sqrt(5)/9 + sqrt(5)/9 + 0 = 17 sqrt(5)/9Compute y_i x_{i+1}:B to C: (0)(2) = 0C to F: (0)(2/9) = 0F to E: (8 sqrt(5)/9)(-1) = -8 sqrt(5)/9E to B: (sqrt(5)/2)(-2) = -sqrt(5)Total: 0 + 0 -8 sqrt(5)/9 -sqrt(5) = -8 sqrt(5)/9 -9 sqrt(5)/9 = -17 sqrt(5)/9Subtract the two sums:17 sqrt(5)/9 - (-17 sqrt(5)/9) = 34 sqrt(5)/9Take absolute value and multiply by 1/2:|34 sqrt(5)/9| * 1/2 = 17 sqrt(5)/9 ≈ 4.223 cm²Same result. So, regardless of the order, I get 17 sqrt(5)/9 cm², which seems too large.But when I split into two triangles, I got sqrt(5) cm², which is about half of ABC's area. That seems more plausible because BEFC is a quadrilateral covering a significant portion but not the entire area.Wait, perhaps the issue is that when I used the shoelace formula for the quadrilateral, I included an area outside the triangle ABC, but since all points are inside ABC, that shouldn't be the case.Alternatively, maybe I made a mistake in the coordinates of point F.Wait, let me recalculate the coordinates of F.We had the circle equation: (x + 1)² + (y - sqrt(5)/2)² = 9/4And the line AC: y = (-sqrt(5)/2)x + sqrt(5)Substituting y into the circle equation:(x + 1)² + [(-sqrt(5)/2 x + sqrt(5)/2)]² = 9/4Expanding:(x² + 2x + 1) + [ (5/4 x² - (sqrt(5)*sqrt(5))/2 x + 5/4 ) ] = 9/4Wait, let me expand [(-sqrt(5)/2 x + sqrt(5)/2)]² correctly.= (sqrt(5)/2 x - sqrt(5)/2)^2= (sqrt(5)/2)^2 (x - 1)^2= (5/4)(x² - 2x + 1)So, the equation becomes:x² + 2x + 1 + (5/4)(x² - 2x + 1) = 9/4Multiply through by 4 to eliminate denominators:4x² + 8x + 4 + 5x² -10x +5 = 9Combine like terms:9x² -2x +9 =9So, 9x² -2x =0x(9x -2)=0x=0 or x=2/9So, x=2/9 is correct, leading to y=8 sqrt(5)/9.So, point F is correctly calculated.Therefore, the issue must be with the shoelace formula. Maybe I should use vectors or another method.Alternatively, perhaps I can find the area of quadrilateral BEFC by subtracting the area of triangle AFE from the area of triangle ABC.But I don't know the area of AFE yet.Alternatively, since E is the midpoint of AB, and the circle centered at E with radius EB intersects AC at F, maybe I can use similar triangles or coordinate geometry to find the ratio AF/FC.Wait, let me think about the coordinates again.Point F is at (2/9, 8 sqrt(5)/9). So, on AC, which goes from A (0, sqrt(5)) to C (2,0). The parametric equation of AC can be written as:x = 2ty = sqrt(5)(1 - t)where t ranges from 0 to 1.At t=0, we are at A (0, sqrt(5)), and at t=1, we are at C (2,0).Point F is at (2/9, 8 sqrt(5)/9). Let's find t such that:2t = 2/9 => t=1/9Check y-coordinate: sqrt(5)(1 - t) = sqrt(5)(1 -1/9)= sqrt(5)(8/9)=8 sqrt(5)/9, which matches.So, t=1/9, meaning AF/AC =1/9, so AF= (1/9)*AC= (1/9)*3=1/3 cm? Wait, no, AC is 3 cm, so AF= (1/9)*3=1/3 cm, and FC=3 -1/3=8/3 cm.Wait, that can't be because AC is 3 cm, so AF=1/3 cm and FC=8/3 cm.But wait, in coordinates, point F is closer to C, so FC should be longer than AF. Indeed, 8/3 cm is longer than 1/3 cm.So, AF=1/3 cm, FC=8/3 cm.Now, the area of triangle ABC is 2 sqrt(5) cm².The area of triangle AFE can be found using the ratio of AF/AC=1/9.Since triangles AFE and ABC are similar (they share angle A and have parallel sides if FE is parallel to BC, but I'm not sure if they are similar).Alternatively, since E is the midpoint of AB, and F divides AC in the ratio AF:FC=1:8, maybe we can use the formula for the area of a triangle with a point dividing sides in certain ratios.Alternatively, use coordinates to find the area of AFE.Point A (0, sqrt(5)), F (2/9, 8 sqrt(5)/9), E (-1, sqrt(5)/2).Using shoelace formula:Coordinates: A (0, sqrt(5)), F (2/9, 8 sqrt(5)/9), E (-1, sqrt(5)/2), back to A.Compute sum of x_i y_{i+1}:0*(8 sqrt(5)/9) + (2/9)*(sqrt(5)/2) + (-1)*sqrt(5) = 0 + (sqrt(5)/9) - sqrt(5) = (sqrt(5)/9 -9 sqrt(5)/9)= -8 sqrt(5)/9Sum of y_i x_{i+1}:sqrt(5)*(2/9) + (8 sqrt(5)/9)*(-1) + (sqrt(5)/2)*0 = (2 sqrt(5)/9) -8 sqrt(5)/9 +0= (-6 sqrt(5)/9)= -2 sqrt(5)/3Subtract the two sums:(-8 sqrt(5)/9) - (-2 sqrt(5)/3)= (-8 sqrt(5)/9 +6 sqrt(5)/9)= (-2 sqrt(5)/9)Take absolute value and multiply by 1/2:| -2 sqrt(5)/9 | *1/2= (2 sqrt(5)/9)*1/2= sqrt(5)/9So, area of triangle AFE is sqrt(5)/9 cm².Therefore, the area of quadrilateral BEFC is area of ABC minus area of AFE: 2 sqrt(5) - sqrt(5)/9= (18 sqrt(5)/9 - sqrt(5)/9)=17 sqrt(5)/9 cm².Wait, that's the same result as the shoelace formula earlier. So, the area of BEFC is indeed 17 sqrt(5)/9 cm².But earlier, when I split BEFC into BEF and EFC, I got sqrt(5) cm², which is approximately 2.236 cm², whereas 17 sqrt(5)/9 ≈4.223 cm². There's a discrepancy here.Wait, perhaps I made a mistake in calculating the area of triangle EFC. Let me recalculate it.Points E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2,0)Using shoelace formula:Coordinates: E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), C (2,0), back to E.Compute sum of x_i y_{i+1}:-1*(8 sqrt(5)/9) + (2/9)*0 + 2*(sqrt(5)/2)= -8 sqrt(5)/9 +0 + sqrt(5)= (-8 sqrt(5)/9 +9 sqrt(5)/9)= sqrt(5)/9Sum of y_i x_{i+1}:(sqrt(5)/2)*(2/9) + (8 sqrt(5)/9)*2 +0*(-1)= (sqrt(5)/9) +16 sqrt(5)/9 +0=17 sqrt(5)/9Subtract the two sums:sqrt(5)/9 -17 sqrt(5)/9= -16 sqrt(5)/9Take absolute value and multiply by 1/2:| -16 sqrt(5)/9 | *1/2=8 sqrt(5)/9So, area of EFC is 8 sqrt(5)/9 cm².Similarly, area of BEF was sqrt(5)/9 cm².Adding them together:8 sqrt(5)/9 + sqrt(5)/9=9 sqrt(5)/9= sqrt(5) cm².But according to the shoelace formula for quadrilateral BEFC, the area is17 sqrt(5)/9 cm².This inconsistency suggests that one of the methods is incorrect. However, since the shoelace formula for the quadrilateral gives 17 sqrt(5)/9, and subtracting the area of AFE from ABC also gives 17 sqrt(5)/9, I think the correct area is17 sqrt(5)/9 cm².The mistake in the earlier method was probably in the calculation of the area of triangle BEF. Let me recalculate that.Points B (-2,0), E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9)Using shoelace formula:Coordinates: B (-2,0), E (-1, sqrt(5)/2), F (2/9, 8 sqrt(5)/9), back to B.Compute sum of x_i y_{i+1}:-2*(sqrt(5)/2) + (-1)*(8 sqrt(5)/9) + (2/9)*0= -sqrt(5) -8 sqrt(5)/9 +0= (-9 sqrt(5)/9 -8 sqrt(5)/9)= -17 sqrt(5)/9Sum of y_i x_{i+1}:0*(-1) + (sqrt(5)/2)*(2/9) + (8 sqrt(5)/9)*(-2)=0 + sqrt(5)/9 -16 sqrt(5)/9= (-15 sqrt(5)/9)= -5 sqrt(5)/3Subtract the two sums:-17 sqrt(5)/9 - (-5 sqrt(5)/3)= -17 sqrt(5)/9 +15 sqrt(5)/9= (-2 sqrt(5)/9)Take absolute value and multiply by 1/2:| -2 sqrt(5)/9 | *1/2= sqrt(5)/9So, area of BEF is sqrt(5)/9 cm².Therefore, the total area of BEFC is area of BEF + area of EFC= sqrt(5)/9 +8 sqrt(5)/9=9 sqrt(5)/9= sqrt(5) cm².But this contradicts the shoelace formula result. Therefore, I must have made a mistake in one of the methods.Wait, perhaps the issue is that when I split BEFC into BEF and EFC, I'm not accounting for overlapping areas or something. Alternatively, maybe the shoelace formula is correct, and the split method is wrong.Given that the shoelace formula for the quadrilateral gives 17 sqrt(5)/9, and subtracting AFE from ABC also gives 17 sqrt(5)/9, I think the correct area is17 sqrt(5)/9 cm².Therefore, the area of quadrilateral BEFC is17 sqrt(5)/9 cm².

answer：Let me try to figure out this problem step by step. So, we have knights in a kingdom, and they're either silver or golden. The problem says that 3/8 of the knights are silver, and the rest are golden. That means the fraction of golden knights is 1 - 3/8, which is 5/8. Next, it mentions that 1/4 of all knights possess shields. So, if we consider the total number of knights as some number, let's say N, then the number of knights with shields is N/4. Now, the tricky part is that the fraction of silver knights who have shields is 3 times the fraction of golden knights who have shields. Let me denote the fraction of golden knights with shields as x. Then, the fraction of silver knights with shields would be 3x. So, if I let the total number of knights be N, then the number of silver knights is (3/8)N, and the number of golden knights is (5/8)N. The number of silver knights with shields would be 3x times the number of silver knights, which is 3x*(3/8)N. Similarly, the number of golden knights with shields would be x times the number of golden knights, which is x*(5/8)N. Since the total number of knights with shields is N/4, I can set up the equation:3x*(3/8)N + x*(5/8)N = (1/4)NSimplifying this equation, I can factor out N:N*(9x/8 + 5x/8) = N/4Combining the terms inside the parentheses:N*(14x/8) = N/4Simplify 14x/8 to 7x/4:N*(7x/4) = N/4Divide both sides by N (assuming N ≠ 0):7x/4 = 1/4Multiply both sides by 4:7x = 1Divide both sides by 7:x = 1/7So, the fraction of golden knights with shields is 1/7, and the fraction of silver knights with shields is 3x, which is 3*(1/7) = 3/7.Therefore, the fraction of silver knights who have shields is 3/7.

answer：Okay, so I have this function here: f(x) = (1/2)^{x² - 2x + 6}. I need to find the interval where this function is monotonically increasing. Hmm, let's see. I remember that for functions like this, which are exponential functions with a base between 0 and 1, their behavior is a bit different compared to exponential functions with a base greater than 1. First, maybe I should analyze the exponent part of the function, which is x² - 2x + 6. That looks like a quadratic function. I think quadratic functions have a parabola shape when graphed, and depending on the coefficient of x², they open upwards or downwards. In this case, the coefficient is positive (1), so the parabola opens upwards. That means it has a minimum point, right?To find the vertex of this parabola, which will give me the minimum point, I can use the formula for the vertex of a quadratic function, which is at x = -b/(2a). Here, a is 1 and b is -2. So, x = -(-2)/(2*1) = 2/2 = 1. So, the vertex is at x = 1. That means the parabola has its minimum at x = 1, and it's symmetric around this line.Now, since the parabola opens upwards, the function x² - 2x + 6 is decreasing on the interval (-∞, 1) and increasing on (1, ∞). Wait, let me make sure: as x approaches 1 from the left, the value of the quadratic decreases, and as x moves away from 1 to the right, it increases. Yeah, that sounds right.So, the exponent part, u(x) = x² - 2x + 6, is decreasing on (-∞, 1) and increasing on (1, ∞). Now, how does this affect the original function f(x) = (1/2)^{u(x)}? I remember that when you have an exponential function with a base between 0 and 1, like (1/2), the function is decreasing. That means if the exponent increases, the overall function decreases, and if the exponent decreases, the overall function increases. So, the behavior of f(x) is opposite to the behavior of u(x).Therefore, since u(x) is decreasing on (-∞, 1), f(x) will be increasing on that interval. Conversely, since u(x) is increasing on (1, ∞), f(x) will be decreasing on that interval. Let me double-check that. If u(x) is decreasing, then (1/2)^{u(x)} is increasing because as u(x) gets smaller, (1/2)^{u(x)} gets larger. For example, if u(x) goes from 10 to 5, (1/2)^{10} is much smaller than (1/2)^5. So yes, f(x) increases as u(x) decreases.So, putting it all together, the function f(x) is increasing when u(x) is decreasing, which is on the interval (-∞, 1). After that point, at x = 1, the function starts decreasing. Wait, should I include x = 1 in the interval? At x = 1, the function f(x) reaches a maximum because that's where u(x) is minimized. So, at x = 1, the function f(x) is at its peak. Therefore, the interval where f(x) is increasing is up to x = 1, but not including x = 1 because at x = 1, it's neither increasing nor decreasing—it's a critical point.So, the monotonic increasing interval is (-∞, 1). I think that's it. Let me just recap:1. Identify the exponent as a quadratic function.2. Find its vertex to determine where it changes from decreasing to increasing.3. Since the base of the exponential function is between 0 and 1, the function f(x) will increase when the exponent decreases and decrease when the exponent increases.4. Therefore, f(x) increases on the interval where the exponent is decreasing, which is (-∞, 1).Yeah, that makes sense. I don't think I missed anything here. It's a good idea to visualize the graph as well. If I imagine the quadratic function u(x), it's a U-shaped parabola with the bottom at x = 1. Then, f(x) is an exponential decay function based on u(x). So, as u(x) goes down to its minimum, f(x) goes up, and as u(x) goes up after x = 1, f(x) goes down. So, the function f(x) is increasing from the left all the way up to x = 1, and then it starts decreasing. Therefore, the interval where it's increasing is all real numbers less than 1. I think I'm confident with this answer now. It's always good to break down the problem into smaller parts and tackle each part step by step. That way, I can make sure I understand each component and how they interact. In this case, understanding the behavior of the quadratic exponent and how it affects the exponential function was key.Another thing to note is the importance of the base of the exponential function. If the base were greater than 1, the behavior would be opposite. For example, if f(x) were 2^{x² - 2x + 6}, then f(x) would increase when u(x) increases and decrease when u(x) decreases. But since our base is 1/2, which is less than 1, the behavior is flipped.This problem is a good example of how the properties of functions, especially composite functions, can affect the overall behavior. It's also a reminder of how critical points, like the vertex of a parabola, can indicate changes in the function's increasing or decreasing nature.I wonder if there's another way to approach this problem, maybe by taking the derivative. Let me try that quickly. If I find the derivative of f(x), I can determine where it's positive (increasing) or negative (decreasing).So, f(x) = (1/2)^{x² - 2x + 6}. Let's take the natural logarithm to make differentiation easier. Let y = f(x), so ln y = (x² - 2x + 6) * ln(1/2). Then, differentiate both sides with respect to x:(1/y) * y' = (2x - 2) * ln(1/2).Therefore, y' = y * (2x - 2) * ln(1/2).Since y = (1/2)^{x² - 2x + 6}, which is always positive, the sign of y' depends on (2x - 2) * ln(1/2).Now, ln(1/2) is negative because 1/2 is less than 1. So, ln(1/2) ≈ -0.693. Therefore, the sign of y' is determined by (2x - 2) multiplied by a negative number.So, y' is positive when (2x - 2) is negative, because a negative times a negative is positive. Similarly, y' is negative when (2x - 2) is positive.Let's solve for when (2x - 2) is negative:2x - 2 < 02x < 2x < 1So, when x < 1, y' is positive, meaning f(x) is increasing. When x > 1, y' is negative, meaning f(x) is decreasing. At x = 1, y' = 0, which is the critical point.This confirms what I found earlier. The derivative method gives the same result, which is reassuring. It's always good to cross-verify using different methods to ensure the answer is correct.So, using both the analysis of the quadratic exponent and the derivative, I've arrived at the same conclusion: the function f(x) is monotonically increasing on the interval (-∞, 1).I think I've covered all the necessary steps and checked my work using two different approaches. I feel confident that this is the correct answer.**Final Answer**The monotonic increasing interval of the function is boxed{(-infty, 1)}.

answer：Alright, so I've got this function here: ( f(x) = frac{x^2 + 1 + a}{x} - a ln x ), where ( a ) is a real number. The problem has two parts. First, when ( a = 1 ), I need to find the extreme values of the function. Second, I have to discuss the monotonicity of the function in general. Hmm, okay, let's tackle part (I) first.Starting with ( a = 1 ), the function simplifies to ( f(x) = frac{x^2 + 2}{x} - ln x ). To find the extreme values, I remember that I need to find the critical points by taking the derivative of ( f(x) ) and setting it equal to zero. So, let's compute ( f'(x) ).First, let's rewrite ( f(x) ) to make differentiation easier. ( f(x) = frac{x^2 + 2}{x} - ln x ) can be rewritten as ( f(x) = x + frac{2}{x} - ln x ). Now, taking the derivative term by term:- The derivative of ( x ) with respect to ( x ) is 1.- The derivative of ( frac{2}{x} ) is ( -frac{2}{x^2} ) using the power rule.- The derivative of ( -ln x ) is ( -frac{1}{x} ).Putting it all together, ( f'(x) = 1 - frac{2}{x^2} - frac{1}{x} ). Hmm, that seems a bit complicated. Maybe I can combine the terms over a common denominator to simplify it. Let's see:( f'(x) = 1 - frac{2}{x^2} - frac{1}{x} = frac{x^2}{x^2} - frac{2}{x^2} - frac{x}{x^2} = frac{x^2 - 2 - x}{x^2} ).So, ( f'(x) = frac{x^2 - x - 2}{x^2} ). Wait, that's a quadratic in the numerator. Maybe I can factor it. Let's try:( x^2 - x - 2 ). Looking for two numbers that multiply to -2 and add to -1. Hmm, 1 and -2? Yes, because 1 * (-2) = -2 and 1 + (-2) = -1. So, factoring, we get ( (x + 1)(x - 2) ).So, ( f'(x) = frac{(x + 1)(x - 2)}{x^2} ). Now, to find critical points, set ( f'(x) = 0 ):( frac{(x + 1)(x - 2)}{x^2} = 0 ). The denominator ( x^2 ) is never zero for ( x > 0 ) (since the domain of ( ln x ) is ( x > 0 )), so we only set the numerator equal to zero:( (x + 1)(x - 2) = 0 ). So, ( x = -1 ) or ( x = 2 ). But since ( x > 0 ), we discard ( x = -1 ) and only consider ( x = 2 ).Okay, so ( x = 2 ) is the critical point. Now, to determine if this is a minimum or maximum, I can use the second derivative test or analyze the sign changes of the first derivative around ( x = 2 ). Let's try the sign analysis.Choose test points around ( x = 2 ). Let's pick ( x = 1 ) and ( x = 3 ).For ( x = 1 ):( f'(1) = frac{(1 + 1)(1 - 2)}{1^2} = frac{2 * (-1)}{1} = -2 ). So, the derivative is negative here.For ( x = 3 ):( f'(3) = frac{(3 + 1)(3 - 2)}{3^2} = frac{4 * 1}{9} = frac{4}{9} ). So, the derivative is positive here.Therefore, the function is decreasing before ( x = 2 ) and increasing after ( x = 2 ). This means ( x = 2 ) is a local minimum.Now, let's find the value of the function at ( x = 2 ):( f(2) = frac{2^2 + 2}{2} - ln 2 = frac{4 + 2}{2} - ln 2 = frac{6}{2} - ln 2 = 3 - ln 2 ).So, the function has a local minimum at ( x = 2 ) with value ( 3 - ln 2 ). Since the function tends to infinity as ( x ) approaches 0 and as ( x ) approaches infinity, this is the only extreme value.Alright, part (I) seems done. Now, moving on to part (II), discussing the monotonicity of ( f(x) ) in general, not just for ( a = 1 ).So, the function is ( f(x) = frac{x^2 + 1 + a}{x} - a ln x ). Let's first find the derivative ( f'(x) ).Rewriting ( f(x) ) as ( f(x) = x + frac{1 + a}{x} - a ln x ). Now, taking the derivative term by term:- The derivative of ( x ) is 1.- The derivative of ( frac{1 + a}{x} ) is ( -frac{1 + a}{x^2} ).- The derivative of ( -a ln x ) is ( -frac{a}{x} ).So, putting it together:( f'(x) = 1 - frac{1 + a}{x^2} - frac{a}{x} ).Let me combine these terms over a common denominator. The common denominator would be ( x^2 ):( f'(x) = frac{x^2}{x^2} - frac{1 + a}{x^2} - frac{a x}{x^2} = frac{x^2 - (1 + a) - a x}{x^2} ).Simplify the numerator:( x^2 - a x - (1 + a) ).So, ( f'(x) = frac{x^2 - a x - (1 + a)}{x^2} ).Hmm, this is a quadratic in the numerator. Let's write it as ( x^2 - a x - (1 + a) ). Maybe factor it? Let's see.Looking for two numbers that multiply to ( -(1 + a) ) and add to ( -a ). Hmm, not sure if it factors nicely. Alternatively, maybe complete the square or find roots.Alternatively, let's set the numerator equal to zero to find critical points:( x^2 - a x - (1 + a) = 0 ).Using the quadratic formula:( x = frac{a pm sqrt{a^2 + 4(1 + a)}}{2} = frac{a pm sqrt{a^2 + 4 + 4a}}{2} = frac{a pm sqrt{(a + 2)^2}}{2} ).Wait, ( a^2 + 4a + 4 = (a + 2)^2 ), so the square root simplifies to ( |a + 2| ). Since ( a ) is real, ( |a + 2| = a + 2 ) if ( a + 2 geq 0 ), else ( -(a + 2) ).So, the roots are:( x = frac{a pm (a + 2)}{2} ).Calculating both possibilities:1. ( x = frac{a + (a + 2)}{2} = frac{2a + 2}{2} = a + 1 ).2. ( x = frac{a - (a + 2)}{2} = frac{-2}{2} = -1 ).Again, since ( x > 0 ), we discard ( x = -1 ) and only consider ( x = a + 1 ).So, the critical point is at ( x = a + 1 ), provided that ( a + 1 > 0 ), which means ( a > -1 ). If ( a leq -1 ), then ( x = a + 1 leq 0 ), which is not in the domain.Therefore, the behavior of ( f'(x) ) depends on the value of ( a ):1. If ( a > -1 ), then ( x = a + 1 ) is a critical point. We can analyze the sign of ( f'(x) ) around this point.2. If ( a leq -1 ), then there are no critical points in the domain ( x > 0 ), so the function is either always increasing or always decreasing.Let's analyze both cases.Case 1: ( a > -1 ).We have a critical point at ( x = a + 1 ). To determine if it's a minimum or maximum, let's check the sign of ( f'(x) ) around ( x = a + 1 ).Choose test points:- For ( x < a + 1 ), say ( x = a ) (assuming ( a > 0 )), plug into ( f'(x) ): ( f'(a) = frac{a^2 - a * a - (1 + a)}{a^2} = frac{a^2 - a^2 - 1 - a}{a^2} = frac{-1 - a}{a^2} ). Since ( a > -1 ), ( -1 - a < 0 ) if ( a > -1 ). So, ( f'(a) < 0 ).- For ( x > a + 1 ), say ( x = a + 2 ), plug into ( f'(x) ): ( f'(a + 2) = frac{(a + 2)^2 - a(a + 2) - (1 + a)}{(a + 2)^2} ). Let's compute the numerator: ( (a^2 + 4a + 4) - (a^2 + 2a) - 1 - a = a^2 + 4a + 4 - a^2 - 2a - 1 - a = (4a - 2a - a) + (4 - 1) = a + 3 ). Since ( a > -1 ), ( a + 3 > 2 ), so the numerator is positive. Therefore, ( f'(a + 2) > 0 ).Thus, the function is decreasing before ( x = a + 1 ) and increasing after ( x = a + 1 ). Therefore, ( x = a + 1 ) is a local minimum.Case 2: ( a leq -1 ).In this case, there are no critical points in ( x > 0 ). To determine the monotonicity, let's analyze the sign of ( f'(x) ).Recall ( f'(x) = frac{x^2 - a x - (1 + a)}{x^2} ).Since ( a leq -1 ), let's see the behavior of the numerator ( x^2 - a x - (1 + a) ).Note that ( x^2 ) is always positive for ( x > 0 ). The term ( -a x ) becomes positive because ( a leq -1 ), so ( -a geq 1 ), and multiplied by ( x > 0 ), it's positive. The term ( -(1 + a) ) is also positive because ( a leq -1 ) implies ( 1 + a leq 0 ), so ( -(1 + a) geq 0 ).Therefore, the numerator is the sum of positive terms: ( x^2 + (-a x) + (-(1 + a)) ), all positive. Hence, the numerator is positive for all ( x > 0 ). Therefore, ( f'(x) > 0 ) for all ( x > 0 ).Thus, when ( a leq -1 ), the function ( f(x) ) is monotonically increasing on its entire domain ( (0, infty) ).Putting it all together:- If ( a > -1 ), ( f(x) ) decreases on ( (0, a + 1) ) and increases on ( (a + 1, infty) ).- If ( a leq -1 ), ( f(x) ) is always increasing on ( (0, infty) ).I think that covers both parts of the problem. Let me just double-check my calculations to make sure I didn't make any mistakes.For part (I), when ( a = 1 ), the critical point was at ( x = 2 ), and the function had a local minimum there. The value was ( 3 - ln 2 ). That seems correct.For part (II), the general case, I found that the critical point is at ( x = a + 1 ) when ( a > -1 ), and the function changes from decreasing to increasing there. When ( a leq -1 ), the function is always increasing. That makes sense because if ( a ) is too negative, the term ( -a ln x ) becomes a large positive term, making the function increase.Yeah, I think that's solid. I don't see any errors in my reasoning.