answer：Okay, so I need to figure out how much Jack will pay back to Jill. Jack borrowed 1200 from Jill and promised to pay back with an interest of 10%. Hmm, okay, so there's a principal amount and an interest rate involved here. First, I should understand what the principal is. The principal is the original amount of money borrowed, which in this case is 1200. The interest rate is 10%, which means Jack has to pay back an additional 10% of the principal amount as interest. So, to find out how much interest Jack has to pay, I can calculate 10% of 1200. To do that, I can convert the percentage to a decimal by dividing by 100, so 10% becomes 0.10. Then, I multiply the principal by this decimal to find the interest.Let me write that down:Interest = Principal × Interest RateInterest = 1200 × 0.10Interest = 120Okay, so the interest Jack needs to pay is 120. Now, to find the total amount Jack has to pay back, I need to add the principal and the interest together.Total Amount = Principal + InterestTotal Amount = 1200 + 120Total Amount = 1320So, Jack will pay back a total of 1320 to Jill. That makes sense because he borrowed 1200 and has to pay back the original amount plus the interest.Wait a minute, let me double-check my calculations to make sure I didn't make any mistakes. If 10% of 1200 is 120, then adding that to the principal gives 1320. Yeah, that seems right. I don't think I missed anything here.Another way to think about it is to consider that paying back with 10% interest means Jack is returning 110% of the principal amount. So, 110% of 1200 is:Total Amount = Principal × (1 + Interest Rate)Total Amount = 1200 × 1.10Total Amount = 1320Yep, that confirms it. Both methods give me the same result, so I'm confident that 1320 is the correct amount Jack needs to pay back to Jill.

answer：Alright, so I've got this problem here about an odd function. Let me try to wrap my head around it step by step. The function f(x) is odd, which I remember means that f(-x) = -f(x) for all x in its domain. Cool, so whatever happens on the positive side of the x-axis, the negative side is just a mirror image flipped over both axes.The problem tells me that when x is greater than 0, f(x) is defined as x squared minus 4x plus 2. So, f(x) = x² - 4x + 2 for x > 0. I need to figure out what f(x) does on the interval from -4 to -2. Specifically, is it increasing or decreasing, and what are its maximum and minimum values there?First, maybe I should analyze the function for x > 0 because that's given, and then use the property of odd functions to find out what happens for x < 0.So, for x > 0, f(x) = x² - 4x + 2. Hmm, that's a quadratic function. Quadratics are parabolas, right? Since the coefficient of x² is positive (1), it opens upwards. That means it has a minimum point at its vertex.To find the vertex, I can use the formula for the x-coordinate of the vertex, which is -b/(2a). In this case, a = 1 and b = -4, so the x-coordinate is -(-4)/(2*1) = 4/2 = 2. So, the vertex is at x = 2. Plugging that back into the function, f(2) = (2)² - 4*(2) + 2 = 4 - 8 + 2 = -2. So, the minimum value of f(x) for x > 0 is -2 at x = 2.Since the parabola opens upwards, the function decreases until x = 2 and then increases after that. So, on the interval [2, 4], which is part of x > 0, the function is increasing. Let me confirm that by taking the derivative. The derivative of f(x) is f'(x) = 2x - 4. At x = 2, the derivative is 0, which is the minimum point. For x > 2, the derivative is positive, meaning the function is increasing.So, on [2, 4], f(x) increases from f(2) = -2 to f(4). Let's calculate f(4): f(4) = 4² - 4*4 + 2 = 16 - 16 + 2 = 2. So, on [2, 4], f(x) goes from -2 to 2, increasing all the way.Now, since f(x) is an odd function, f(-x) = -f(x). That means whatever happens on the positive side is mirrored and flipped on the negative side. So, if f(x) is increasing on [2, 4], then f(-x) should be increasing on [-4, -2]. Wait, is that right?Let me think. If f(x) is increasing on [2, 4], then f(-x) = -f(x). So, as x increases from 2 to 4, f(x) increases from -2 to 2. Therefore, as x decreases from -2 to -4 (which is the same as increasing from 2 to 4 in the negative direction), f(-x) = -f(x) would go from -(-2) = 2 to -(2) = -2. So, f(x) on [-4, -2] goes from -2 to 2 as x goes from -4 to -2. Wait, that seems like it's increasing as well.But hold on, when x increases from -4 to -2, what does f(x) do? Let's see. If x increases from -4 to -2, that's like moving from left to right on the negative side. Since f(x) is odd, f(-4) = -f(4) = -2, and f(-2) = -f(2) = 2. So, as x increases from -4 to -2, f(x) increases from -2 to 2. So, it's increasing on [-4, -2].But wait, the options talk about maximum and minimum values. On [-4, -2], the function f(x) starts at f(-4) = -2 and goes up to f(-2) = 2. So, the maximum value is 2, and the minimum is -2. So, it's increasing with a maximum of 2 and a minimum of -2.Looking back at the options:A: monotonically increasing with a maximum value of 2B: monotonically increasing with a minimum value of 2C: monotonically decreasing with a maximum value of -2D: monotonically decreasing with a minimum value of -2So, A says it's increasing with a max of 2, which matches what I found. B says increasing with a min of 2, which is not correct because the min is -2. C and D say it's decreasing, which contradicts my conclusion that it's increasing.Therefore, the correct answer should be A.Wait, but let me double-check. Maybe I messed up the direction. If f(x) is increasing on [2,4], then f(-x) = -f(x). So, as x increases from -4 to -2, f(x) = -f(-x). Since f(-x) is increasing from f(-4) = -2 to f(-2) = 2, then f(x) = -f(-x) would be decreasing from 2 to -2. Wait, that's conflicting with my earlier conclusion.Hold on, maybe I need to clarify this. Let's take specific points. For x = 2, f(2) = -2. For x = 4, f(4) = 2. So, on [2,4], f(x) increases from -2 to 2.Now, for x = -2, f(-2) = -f(2) = -(-2) = 2. For x = -4, f(-4) = -f(4) = -2. So, on [-4, -2], f(x) goes from -2 at x = -4 to 2 at x = -2. So, as x increases from -4 to -2, f(x) increases from -2 to 2. Therefore, it's increasing on that interval.But earlier, when I thought about f(-x) = -f(x), I might have confused myself. Let me think again. If f(x) is increasing on [2,4], then for x in [2,4], as x increases, f(x) increases. For x in [-4,-2], as x increases (from -4 to -2), f(x) = -f(-x). Since x is increasing, -x is decreasing from 4 to 2. So, f(-x) is decreasing from 2 to -2, hence f(x) = -f(-x) is increasing from -2 to 2. So, yes, f(x) is increasing on [-4,-2].Therefore, the function is monotonically increasing on [-4,-2] with a maximum value of 2 at x = -2 and a minimum value of -2 at x = -4. So, the correct option is A.I think I got confused earlier when trying to relate f(-x) and f(x), but breaking it down point by point clarified it. So, yeah, A is the right answer.

answer：To find ( f(x+3) ) for the function ( f(x) = frac{x(x-1)}{2} ), we substitute ( x+3 ) into the function:[f(x+3) = frac{(x+3)(x+3-1)}{2} = frac{(x+3)(x+2)}{2}]Next, we expand the numerator:[(x+3)(x+2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6]Thus, the function simplifies to:[f(x+3) = frac{x^2 + 5x + 6}{2}]Therefore, the correct answer is (boxed{textbf{(B)}}).

answer：Okay, so Wayne wants to serve shrimp cocktail as an appetizer for his guests. He's planning on giving each guest 5 shrimp, and he's expecting 40 guests. That means I need to figure out how many shrimp he needs in total. Let me do that first.So, if there are 40 guests and each guest gets 5 shrimp, then the total number of shrimp needed is 40 multiplied by 5. Let me write that down: 40 times 5 equals 200 shrimp. Okay, so Wayne needs 200 shrimp in total.Now, the next part is about the cost. The problem says that the shrimp costs a certain amount per pound, and each pound has 20 shrimp. So, I need to figure out how many pounds of shrimp Wayne needs to buy to get 200 shrimp.If each pound has 20 shrimp, then to find out how many pounds are needed for 200 shrimp, I can divide the total number of shrimp by the number of shrimp per pound. So, 200 divided by 20 equals 10 pounds. Got it, Wayne needs to buy 10 pounds of shrimp.The problem also mentions that Wayne spends 170 on the appetizer. I assume that this 170 is the total cost for the shrimp. So, if 10 pounds of shrimp cost 170, I can find out how much one pound of shrimp costs by dividing the total cost by the number of pounds.Let me do that calculation: 170 divided by 10 pounds equals 17 per pound. So, the shrimp costs 17 per pound.Wait, let me double-check my calculations to make sure I didn't make any mistakes. First, 40 guests times 5 shrimp per guest is indeed 200 shrimp. Then, 200 shrimp divided by 20 shrimp per pound gives 10 pounds. And finally, 170 divided by 10 pounds is 17 per pound. Yeah, that seems correct.I think I've got it. The shrimp costs 17 per pound.

answer：To determine the largest two-digit integer ( n ) that is the product of two distinct factors ( x ) and ( 10x + 2x ), where ( x ) is a single-digit integer, we start by expressing ( n ) as:[n = x times (10x + 2x) = x times 12x = 12x^2]Next, we need to find the largest ( x ) such that ( 12x^2 ) remains a two-digit number. We evaluate ( 12x^2 ) for ( x ) from 1 to 9:- For ( x = 1 ): ( 12 times 1^2 = 12 )- For ( x = 2 ): ( 12 times 2^2 = 48 )- For ( x = 3 ): ( 12 times 3^2 = 108 ) (exceeds two digits)The largest valid ( x ) that keeps ( n ) as a two-digit number is ( x = 2 ). Therefore, the largest possible value of ( n ) is:[n = 12 times 2^2 = 48]

answer：Okay, so I need to find the determinant of this matrix:[begin{vmatrix} -5 & 8 3 & -4 end{vmatrix}]Hmm, determinants. I remember they're a key part of matrices, especially for 2x2 matrices. But wait, what exactly is a determinant? I think it's a scalar value that can be computed from the elements of a square matrix and it gives some information about the matrix, like whether it's invertible or not. But I'm not entirely sure about all the details.Alright, focusing on the task at hand. I need to compute the determinant of this specific 2x2 matrix. I recall there's a formula for the determinant of a 2x2 matrix. Let me try to remember it. I think it's something like multiplying the top-left element by the bottom-right element and then subtracting the product of the top-right and bottom-left elements. So, if the matrix is:[begin{vmatrix} a & b c & d end{vmatrix}]Then the determinant is ( ad - bc ). Yeah, that sounds right. Let me double-check that. I think that's correct because I remember seeing it in my notes. So, applying this formula to the given matrix.First, identify the elements:- ( a = -5 ) (top-left)- ( b = 8 ) (top-right)- ( c = 3 ) (bottom-left)- ( d = -4 ) (bottom-right)So, according to the formula, the determinant should be:[text{det} = (-5)(-4) - (8)(3)]Let me compute each part step by step to avoid mistakes. First, multiply ( a ) and ( d ):[(-5) times (-4) = 20]Okay, that's straightforward. Multiplying two negative numbers gives a positive result.Next, multiply ( b ) and ( c ):[8 times 3 = 24]That's also straightforward. Now, subtract the second product from the first:[20 - 24 = -4]So, the determinant is (-4). Wait, let me make sure I didn't make any calculation errors. It's easy to mix up signs or multiplication. Let me go through it again.First product: ( (-5) times (-4) = 20 ). Yep, that's correct because a negative times a negative is a positive.Second product: ( 8 times 3 = 24 ). That's also correct.Subtracting: ( 20 - 24 = -4 ). Yep, that's right. So, the determinant is indeed (-4).I think I've got it. The determinant of the matrix is (-4). I don't see any mistakes in my calculations, so I'm confident with this answer.