answer：Okay, so I have this problem here: Let f(x) = x² - 6x + 14 and g(f(x)) = 3x + 4. I need to find the sum of all possible values of g(7). Hmm, okay. Let me try to figure this out step by step.First, I know that f(x) is a quadratic function, and g is a function that takes f(x) as its input. So, g(f(x)) is essentially a composition of functions. The problem is asking for g(7), which means I need to find the value of g when its input is 7. But since g is defined in terms of f(x), I think I need to find the x values that make f(x) equal to 7, and then plug those x values into g(f(x)) to get the corresponding g(7) values. Then, I can sum those up.So, let's start by setting f(x) equal to 7 and solving for x. That should give me the x values that correspond to f(x) = 7.f(x) = x² - 6x + 14 = 7Subtracting 7 from both sides to set the equation to zero:x² - 6x + 14 - 7 = 0 x² - 6x + 7 = 0Okay, so now I have a quadratic equation: x² - 6x + 7 = 0. I need to solve for x. I can use the quadratic formula here, which is x = [-b ± √(b² - 4ac)] / (2a). In this equation, a = 1, b = -6, and c = 7.Plugging those values into the quadratic formula:x = [6 ± √((-6)² - 4*1*7)] / (2*1) x = [6 ± √(36 - 28)] / 2 x = [6 ± √8] / 2Simplifying √8, which is 2√2, so:x = [6 ± 2√2] / 2I can factor out a 2 from the numerator:x = 2[3 ± √2] / 2 x = 3 ± √2So, the solutions are x = 3 + √2 and x = 3 - √2. These are the two x values that make f(x) equal to 7.Now, since g(f(x)) = 3x + 4, and we have f(x) = 7 for these x values, we can substitute each x into g(f(x)) to find the corresponding g(7) values.Let's compute g(7) for x = 3 + √2 first:g(f(3 + √2)) = 3*(3 + √2) + 4 = 9 + 3√2 + 4 = (9 + 4) + 3√2 = 13 + 3√2Now, let's compute g(7) for x = 3 - √2:g(f(3 - √2)) = 3*(3 - √2) + 4 = 9 - 3√2 + 4 = (9 + 4) - 3√2 = 13 - 3√2So, the two possible values of g(7) are 13 + 3√2 and 13 - 3√2.The problem asks for the sum of all possible values of g(7). So, I need to add these two results together:(13 + 3√2) + (13 - 3√2) = 13 + 13 + 3√2 - 3√2 = 26 + 0 = 26Wait, that's interesting. The √2 terms cancel each other out, leaving just 26. So, the sum of all possible values of g(7) is 26.Let me double-check my steps to make sure I didn't make a mistake. First, solving f(x) = 7 gave me x = 3 ± √2. Then, plugging those into g(f(x)) = 3x + 4 gave me 13 ± 3√2. Adding those together cancels the radical terms, resulting in 26. That seems correct.Alternatively, is there another way to approach this problem without finding the specific x values? Maybe by considering the relationship between f(x) and g(f(x)).Since g(f(x)) = 3x + 4, perhaps I can express x in terms of f(x) and then substitute f(x) = 7. Let's see.From f(x) = x² - 6x + 14, I can write this as:x² - 6x + 14 = f(x)But I need to solve for x in terms of f(x). That might be complicated because it's a quadratic equation. Alternatively, maybe I can find a relationship between x and f(x) that allows me to express x as a function of f(x), but that might not be straightforward.Alternatively, since g(f(x)) = 3x + 4, if I let y = f(x), then g(y) = 3x + 4. But since y = f(x), which is x² - 6x + 14, I can write x in terms of y. However, solving for x in terms of y would require solving the quadratic equation, which is what I did earlier, leading back to the same x values.So, perhaps the method I used initially is the most straightforward. It gives me two x values, each leading to a specific g(7) value, and their sum is 26.I think that's solid. I don't see any errors in my calculations, and the logic flows correctly. So, I'm confident that the sum of all possible values of g(7) is 26.

answer：Okay, so I have this problem where I need to find the area bounded by the graphs of ( f(x) ) and its inverse ( g(x) ), along with a line segment connecting the points ( (3, 2) ) and ( (2, 3) ). Let me try to break this down step by step.First, I know that ( f(x) ) is an increasing continuous function defined on the interval ([0, 3]). It starts at ( f(0) = 0 ) and ends at ( f(3) = 2 ). The area under ( f(x) ) from 0 to 3 is given as 2. Since ( g(x) ) is the inverse of ( f(x) ), it means that ( g(f(x)) = x ) and ( f(g(x)) = x ). Also, it's given that ( g(x) > f(x) ) for all positive ( x ) where both are defined. So, ( g(x) ) is above ( f(x) ) in their overlapping domain.I need to find the area bounded by ( f(x) ), ( g(x) ), and the line segment connecting ( (3, 2) ) and ( (2, 3) ). Let me visualize this. The function ( f(x) ) goes from ( (0, 0) ) to ( (3, 2) ), and its inverse ( g(x) ) goes from ( (0, 0) ) to ( (2, 3) ). The line segment connects ( (3, 2) ) to ( (2, 3) ), forming a triangle-like shape with the two functions.I remember that the area between a function and its inverse can be found using the integral of ( f(x) ) and the integral of ( g(x) ). Since ( g(x) ) is the inverse of ( f(x) ), the integral of ( g(x) ) from ( a ) to ( b ) is equal to ( b cdot g(b) - a cdot g(a) - int_{g(a)}^{g(b)} f(x) , dx ). But I need to be careful with the limits here.Given that ( f(x) ) is defined on ([0, 3]) and ( f(3) = 2 ), the inverse function ( g(x) ) will be defined on ([0, 2]) and ( g(2) = 3 ). So, the area between ( f(x) ) and ( g(x) ) from 0 to 2 would involve integrating both functions over their respective domains.But wait, the problem also mentions the line segment connecting ( (3, 2) ) and ( (2, 3) ). So, I think the region we're interested in is bounded by ( f(x) ) from ( x = 0 ) to ( x = 3 ), ( g(x) ) from ( x = 0 ) to ( x = 2 ), and the line segment from ( (3, 2) ) to ( (2, 3) ).Let me try to sketch this mentally. The graph of ( f(x) ) starts at the origin and goes up to ( (3, 2) ). The inverse function ( g(x) ) starts at the origin and goes up to ( (2, 3) ). The line segment connects ( (3, 2) ) to ( (2, 3) ), forming a sort of closed region with the two curves.I think the area we need is the area between ( f(x) ) and ( g(x) ) from 0 to 2, plus the area under the line segment from ( x = 2 ) to ( x = 3 ). But I'm not entirely sure. Maybe it's better to calculate the area bounded by all three: ( f(x) ), ( g(x) ), and the line segment.Let me recall that the area between ( f(x) ) and its inverse ( g(x) ) from ( x = a ) to ( x = b ) can be found by integrating ( f(x) ) and ( g(x) ) and subtracting appropriately. But since ( g(x) ) is above ( f(x) ), the area between them would be the integral of ( g(x) - f(x) ) over their overlapping domain.However, in this case, the overlapping domain where both ( f(x) ) and ( g(x) ) are defined is from ( x = 0 ) to ( x = 2 ), because ( f(x) ) is defined up to ( x = 3 ), but ( g(x) ) is only defined up to ( x = 2 ). So, the area between ( f(x) ) and ( g(x) ) from 0 to 2 would be ( int_{0}^{2} (g(x) - f(x)) , dx ).But then, beyond ( x = 2 ), ( g(x) ) is not defined, but ( f(x) ) continues up to ( x = 3 ). However, the region is also bounded by the line segment connecting ( (3, 2) ) and ( (2, 3) ). So, I think the total area is the sum of two parts:1. The area between ( f(x) ) and ( g(x) ) from ( x = 0 ) to ( x = 2 ).2. The area between ( f(x) ) and the line segment from ( x = 2 ) to ( x = 3 ).Alternatively, maybe it's the area bounded by ( f(x) ), ( g(x) ), and the line segment, which forms a closed region. To find this, I might need to consider the area under ( f(x) ) from 0 to 3, the area under ( g(x) ) from 0 to 2, and subtract the area under the line segment.Wait, let me think again. The area under ( f(x) ) from 0 to 3 is given as 2. The area under ( g(x) ) from 0 to 2 can be found using the fact that ( g(x) ) is the inverse of ( f(x) ). There's a property that the integral of ( f(x) ) from ( a ) to ( b ) plus the integral of ( g(x) ) from ( f(a) ) to ( f(b) ) equals ( b cdot f(b) - a cdot f(a) ). So, in this case, ( int_{0}^{3} f(x) , dx + int_{0}^{2} g(x) , dx = 3 cdot 2 - 0 cdot 0 = 6 ). Since ( int_{0}^{3} f(x) , dx = 2 ), then ( int_{0}^{2} g(x) , dx = 6 - 2 = 4 ).So, the area under ( g(x) ) from 0 to 2 is 4. Now, the area between ( f(x) ) and ( g(x) ) from 0 to 2 would be ( int_{0}^{2} (g(x) - f(x)) , dx ). But I don't know ( f(x) ) explicitly, so maybe I can express this in terms of the given integrals.Wait, ( int_{0}^{2} g(x) , dx = 4 ) and ( int_{0}^{3} f(x) , dx = 2 ). But ( f(x) ) is defined on [0,3], while ( g(x) ) is defined on [0,2]. So, to find the area between them, I need to consider the overlapping domain, which is [0,2]. So, the area between ( f(x) ) and ( g(x) ) from 0 to 2 is ( int_{0}^{2} (g(x) - f(x)) , dx ).But I don't know ( f(x) ) explicitly, so maybe I can use the relationship between ( f(x) ) and ( g(x) ). Since ( g(x) ) is the inverse of ( f(x) ), we have ( f(g(x)) = x ) and ( g(f(x)) = x ). Also, the area under ( f(x) ) from 0 to 3 is 2, and the area under ( g(x) ) from 0 to 2 is 4.But how does this help me find the area between ( f(x) ) and ( g(x) )? Maybe I can think of the area between them as the area under ( g(x) ) minus the area under ( f(x) ) over the interval where both are defined, which is [0,2]. So, ( int_{0}^{2} g(x) , dx - int_{0}^{2} f(x) , dx ).But I know ( int_{0}^{2} g(x) , dx = 4 ) and ( int_{0}^{3} f(x) , dx = 2 ). However, ( int_{0}^{2} f(x) , dx ) is part of the total area under ( f(x) ). Let me denote ( A = int_{0}^{2} f(x) , dx ) and ( B = int_{2}^{3} f(x) , dx ). So, ( A + B = 2 ).But I don't know ( A ) or ( B ) individually. Hmm, maybe I need another approach.Alternatively, I can consider the area bounded by ( f(x) ), ( g(x) ), and the line segment as the sum of two areas:1. The area between ( f(x) ) and ( g(x) ) from 0 to 2.2. The area between ( f(x) ) and the line segment from 2 to 3.But to find the area between ( f(x) ) and the line segment, I need the equation of the line segment connecting ( (3, 2) ) and ( (2, 3) ).Let me find the equation of the line segment. The two points are ( (3, 2) ) and ( (2, 3) ). The slope ( m ) is ( (3 - 2)/(2 - 3) = 1/(-1) = -1 ). So, the slope is -1. Using point-slope form with point ( (3, 2) ):( y - 2 = -1(x - 3) )( y = -x + 5 )So, the equation of the line is ( y = -x + 5 ). This line is valid from ( x = 2 ) to ( x = 3 ).Now, the area between ( f(x) ) and this line from ( x = 2 ) to ( x = 3 ) would be ( int_{2}^{3} [(-x + 5) - f(x)] , dx ).But again, I don't know ( f(x) ) explicitly, so I need another way to express this integral.Wait, maybe I can relate the areas using the inverse function properties. Since ( g(x) ) is the inverse of ( f(x) ), the area under ( g(x) ) from 0 to 2 is 4, as I found earlier. The area under ( f(x) ) from 0 to 3 is 2. The area under the line segment from ( x = 2 ) to ( x = 3 ) can be found as the area of the triangle formed by the points ( (2, 3) ), ( (3, 2) ), and ( (2, 2) ) or something like that.Wait, actually, the line segment is from ( (3, 2) ) to ( (2, 3) ), so it's a straight line with slope -1. The area under this line from ( x = 2 ) to ( x = 3 ) is a trapezoid or a triangle. Let me calculate it.The area under the line ( y = -x + 5 ) from ( x = 2 ) to ( x = 3 ) is the integral ( int_{2}^{3} (-x + 5) , dx ).Calculating this integral:( int_{2}^{3} (-x + 5) , dx = left[ -frac{1}{2}x^2 + 5x right]_{2}^{3} )At ( x = 3 ):( -frac{1}{2}(9) + 5(3) = -4.5 + 15 = 10.5 )At ( x = 2 ):( -frac{1}{2}(4) + 5(2) = -2 + 10 = 8 )So, the integral is ( 10.5 - 8 = 2.5 ).So, the area under the line segment from ( x = 2 ) to ( x = 3 ) is 2.5.Now, the total area bounded by ( f(x) ), ( g(x) ), and the line segment would be the sum of the area between ( f(x) ) and ( g(x) ) from 0 to 2 and the area between ( f(x) ) and the line segment from 2 to 3.But I still need to find the area between ( f(x) ) and ( g(x) ) from 0 to 2. As I mentioned earlier, ( int_{0}^{2} g(x) , dx = 4 ) and ( int_{0}^{3} f(x) , dx = 2 ). Let me denote ( A = int_{0}^{2} f(x) , dx ) and ( B = int_{2}^{3} f(x) , dx ), so ( A + B = 2 ).The area between ( f(x) ) and ( g(x) ) from 0 to 2 is ( int_{0}^{2} (g(x) - f(x)) , dx = int_{0}^{2} g(x) , dx - int_{0}^{2} f(x) , dx = 4 - A ).But I don't know ( A ). However, I can relate ( A ) and ( B ) using the fact that ( f(x) ) is increasing and ( g(x) ) is its inverse. There's a property that the area under ( f(x) ) from ( a ) to ( b ) plus the area under ( g(x) ) from ( f(a) ) to ( f(b) ) equals ( b cdot f(b) - a cdot f(a) ). So, applying this to ( a = 0 ) and ( b = 3 ):( int_{0}^{3} f(x) , dx + int_{0}^{2} g(x) , dx = 3 cdot 2 - 0 cdot 0 = 6 )We know ( int_{0}^{3} f(x) , dx = 2 ) and ( int_{0}^{2} g(x) , dx = 4 ), which checks out because ( 2 + 4 = 6 ).But how does this help me find ( A )? Let me consider the area under ( f(x) ) from 0 to 2, which is ( A ), and the area under ( g(x) ) from 0 to 2, which is 4. The area between ( f(x) ) and ( g(x) ) from 0 to 2 is ( 4 - A ).But I also know that ( A + B = 2 ), so ( B = 2 - A ). The area between ( f(x) ) and the line segment from 2 to 3 is ( int_{2}^{3} [(-x + 5) - f(x)] , dx = int_{2}^{3} (-x + 5) , dx - int_{2}^{3} f(x) , dx = 2.5 - B ).So, the total area bounded by all three is ( (4 - A) + (2.5 - B) ). But since ( A + B = 2 ), this becomes ( 4 - A + 2.5 - B = 6.5 - (A + B) = 6.5 - 2 = 4.5 ).Wait, that seems too straightforward. Let me verify.Total area = area between ( f(x) ) and ( g(x) ) from 0 to 2 + area between ( f(x) ) and the line segment from 2 to 3.Area between ( f(x) ) and ( g(x) ) from 0 to 2 = ( int_{0}^{2} (g(x) - f(x)) , dx = 4 - A ).Area between ( f(x) ) and the line segment from 2 to 3 = ( int_{2}^{3} [(-x + 5) - f(x)] , dx = 2.5 - B ).Total area = ( (4 - A) + (2.5 - B) = 6.5 - (A + B) = 6.5 - 2 = 4.5 ).Yes, that makes sense. So, the total area bounded by ( f(x) ), ( g(x) ), and the line segment is 4.5.I think that's the answer. Let me just recap to make sure I didn't miss anything.1. Found the equation of the line segment: ( y = -x + 5 ).2. Calculated the area under this line from ( x = 2 ) to ( x = 3 ): 2.5.3. Used the property of inverse functions to find the area under ( g(x) ) from 0 to 2: 4.4. Noted that the area under ( f(x) ) from 0 to 3 is 2, so the area under ( f(x) ) from 0 to 2 is ( A ) and from 2 to 3 is ( B ), with ( A + B = 2 ).5. Expressed the total bounded area as ( (4 - A) + (2.5 - B) = 6.5 - (A + B) = 4.5 ).Yes, that seems correct. I think the area is 4.5.

answer：First, I need to determine the total cost of the car after repairs. This includes the initial purchase price plus the repair costs.Next, I'll use the selling price and the profit percentage to set up an equation. The profit can be expressed both in terms of the selling price and the total cost.By equating the two expressions for profit, I can solve for the repair costs. This involves rearranging the equation to isolate the repair costs and then performing the necessary calculations to find the exact amount spent on repairs.

answer：Okay, so I have this problem here: I need to find the value of x that gives the minimum value for the expression x squared plus 14x minus 12. Hmm, let's see. I remember that quadratic expressions have either a minimum or a maximum value, depending on the coefficient of the x squared term. In this case, the coefficient is 1, which is positive, so the parabola opens upwards, meaning it has a minimum point. That makes sense because the problem is asking for the minimum value.Alright, so how do I find the value of x that gives this minimum? I think there are a couple of methods. One is completing the square, and another is using the vertex formula. I'm not entirely sure which one is easier, but maybe I'll try both to see which one clicks better.First, let's try completing the square. I remember that completing the square involves turning the quadratic expression into a perfect square trinomial plus some constant. So, starting with the expression x squared plus 14x minus 12. The first two terms are x squared and 14x. To complete the square, I need to take half of the coefficient of x, which is 14, so half of 14 is 7. Then, I square that, so 7 squared is 49. So, I can rewrite the expression as (x squared plus 14x plus 49) minus 49 minus 12. Wait, why minus 49? Because I added 49 to complete the square, but I need to subtract it to keep the expression equivalent to the original one. So, that becomes (x + 7) squared minus 49 minus 12. Let's compute that: 49 plus 12 is 61, so it's (x + 7) squared minus 61.Now, since (x + 7) squared is a square term, it's always non-negative, right? The smallest value it can take is zero. So, the minimum value of the entire expression occurs when (x + 7) squared is zero. That happens when x + 7 equals zero, which means x equals -7. So, the value of x that gives the minimum is -7.Wait, let me double-check that. If I plug x = -7 back into the original expression, does it give me the minimum value? Let's see: (-7) squared is 49, plus 14 times -7 is -98, minus 12. So, 49 minus 98 is -49, minus 12 is -61. That matches the constant term after completing the square, so it seems correct.Alternatively, I could use the vertex formula. For a quadratic equation in the form ax squared plus bx plus c, the x-coordinate of the vertex is given by -b over 2a. In this case, a is 1 and b is 14. So, plugging in, x equals -14 divided by 2 times 1, which is -14 over 2, which simplifies to -7. So, that confirms the result I got from completing the square.Hmm, both methods give me the same answer, so I'm pretty confident that x equals -7 is the correct value that gives the minimum value for the expression.Just to make sure I didn't make any mistakes, let me think through the steps again. Completing the square: take x squared plus 14x, add and subtract (14/2)^2 which is 49, so x squared plus 14x plus 49 minus 49 minus 12. That becomes (x + 7)^2 minus 61. The minimum occurs when the square term is zero, so x + 7 = 0, hence x = -7.Using the vertex formula: x = -b/(2a) = -14/(2*1) = -7. Yep, that checks out.I think I'm good here. The answer is x = -7.

answer：To determine the total distance Margo walked, we'll start by calculating the total time she spent walking. She walked for 15 minutes to her friend's house and 25 minutes back home. First, add the two times together:[15 text{ minutes} + 25 text{ minutes} = 40 text{ minutes}]Next, convert the total time from minutes to hours:[40 text{ minutes} = frac{40}{60} text{ hours} = frac{2}{3} text{ hours}]Now, use the formula for distance, which is the product of rate and time. Margo's average walking rate is 3 miles per hour:[text{distance} = text{rate} times text{time} = 3 text{ miles/hour} times frac{2}{3} text{ hours} = 2 text{ miles}]Therefore, Margo walked a total of (boxed{2}) miles.

answer：Okay, so I have this problem about a quadratic function f(x) = ax² + bx, where a isn't zero, and some conditions are given. I need to find the expression for f(x) and then find the maximum value of another function g(x) on the interval [0,3]. Let me try to break this down step by step.First, part (I): Find the analytical expression of f(x). The function is quadratic, so it's a parabola. The first condition is that f(1 - x) = f(1 + x). Hmm, that seems like it's saying that the function has some symmetry around x = 1. Because if I plug in 1 - x and 1 + x, they give the same value, which usually means the axis of symmetry is at x = 1. For a quadratic function ax² + bx + c, the axis of symmetry is at x = -b/(2a). But in this case, the function is given as f(x) = ax² + bx, so c is zero. So the axis of symmetry is x = -b/(2a). According to the condition, this should be equal to 1. So I can write:-b/(2a) = 1That's equation (1). So that's one condition relating a and b.The second condition is that the equation f(x) = 2x has two equal real roots. So let's write that equation:ax² + bx = 2xSubtract 2x from both sides:ax² + (b - 2)x = 0This is a quadratic equation. It has two equal real roots, which means the discriminant is zero. For a quadratic equation ax² + bx + c = 0, the discriminant is b² - 4ac. In this case, the equation is ax² + (b - 2)x + 0 = 0, so the discriminant is (b - 2)² - 4*a*0 = (b - 2)². Since the discriminant is zero, we have:(b - 2)² = 0Taking the square root of both sides:b - 2 = 0So, b = 2.Now, substitute b = 2 into equation (1):-2/(2a) = 1Simplify numerator:-2/(2a) = -1/a = 1So, -1/a = 1 implies that a = -1.Therefore, the quadratic function is f(x) = -x² + 2x.Wait, let me double-check that. If a = -1 and b = 2, then f(x) = -x² + 2x. Let's verify the conditions.First condition: f(1 - x) = f(1 + x). Let's compute f(1 - x):f(1 - x) = -(1 - x)² + 2(1 - x) = -(1 - 2x + x²) + 2 - 2x = -1 + 2x - x² + 2 - 2x = (-1 + 2) + (2x - 2x) + (-x²) = 1 - x².Similarly, f(1 + x) = -(1 + x)² + 2(1 + x) = -(1 + 2x + x²) + 2 + 2x = -1 - 2x - x² + 2 + 2x = (-1 + 2) + (-2x + 2x) + (-x²) = 1 - x².So yes, f(1 - x) = f(1 + x) = 1 - x². That checks out.Second condition: f(x) = 2x has two equal real roots. Let's solve f(x) = 2x:- x² + 2x = 2xSubtract 2x from both sides:- x² = 0So x² = 0, which gives x = 0 as a double root. So that's correct, it has two equal real roots at x = 0.Alright, so part (I) is done. f(x) = -x² + 2x.Now, part (II): Find the maximum value of g(x) on [0, 3], where g(x) is defined as (1/3)x³ - x - f(x). Since we found f(x), let's substitute that in.g(x) = (1/3)x³ - x - (-x² + 2x) = (1/3)x³ - x + x² - 2x = (1/3)x³ + x² - 3x.So, g(x) = (1/3)x³ + x² - 3x.We need to find the maximum value of this function on the interval [0, 3]. To find extrema on a closed interval, we can use the Extreme Value Theorem, which says that continuous functions on closed intervals attain their maxima and minima either at critical points or at the endpoints.So, first, let's find the critical points by taking the derivative of g(x) and setting it equal to zero.g'(x) = d/dx [(1/3)x³ + x² - 3x] = x² + 2x - 3.Set g'(x) = 0:x² + 2x - 3 = 0This is a quadratic equation. Let's solve for x.Using quadratic formula: x = [-b ± sqrt(b² - 4ac)]/(2a)Here, a = 1, b = 2, c = -3.Discriminant: b² - 4ac = 4 - 4*1*(-3) = 4 + 12 = 16So, sqrt(16) = 4Thus, x = [-2 ± 4]/2So, two solutions:x = (-2 + 4)/2 = 2/2 = 1x = (-2 - 4)/2 = -6/2 = -3So, critical points at x = 1 and x = -3.But our interval is [0, 3], so x = -3 is outside the interval. Therefore, the only critical point inside [0, 3] is x = 1.Therefore, the possible points where the maximum can occur are x = 0, x = 1, and x = 3.Now, let's compute g(x) at these points.First, x = 0:g(0) = (1/3)(0)³ + (0)² - 3*(0) = 0 + 0 - 0 = 0.Next, x = 1:g(1) = (1/3)(1)³ + (1)² - 3*(1) = (1/3) + 1 - 3 = (1/3) + (3/3) - (9/3) = (4/3) - (9/3) = (-5/3).Then, x = 3:g(3) = (1/3)(27) + (9) - 9 = 9 + 9 - 9 = 9.So, g(0) = 0, g(1) = -5/3, and g(3) = 9.Therefore, among these, the maximum value is 9 at x = 3.Wait, but just to be thorough, let's make sure that there are no other critical points in the interval. We found x = 1 and x = -3, but x = -3 is outside [0,3], so x = 1 is the only critical point. So, we evaluated all critical points and endpoints, and the maximum is indeed at x = 3.Just to visualize, let's think about the behavior of g(x). The derivative g'(x) = x² + 2x - 3. Let's analyze the sign of g'(x) on [0,3].We can factor g'(x) as (x + 3)(x - 1). So, the critical points are x = -3 and x = 1. On the interval [0,3], let's pick test points in the intervals [0,1) and (1,3].For x in [0,1), let's choose x = 0.5:g'(0.5) = (0.5 + 3)(0.5 - 1) = (3.5)(-0.5) = -1.75 < 0. So, g(x) is decreasing on [0,1).For x in (1,3], let's choose x = 2:g'(2) = (2 + 3)(2 - 1) = (5)(1) = 5 > 0. So, g(x) is increasing on (1,3].Therefore, on [0,1], g(x) is decreasing, reaching a minimum at x = 1, then increasing from x = 1 to x = 3. Hence, the maximum on [0,3] must be at one of the endpoints, either x = 0 or x = 3. Since g(0) = 0 and g(3) = 9, the maximum is at x = 3.Therefore, the maximum value of g(x) on [0,3] is 9.I think that's solid. Let me just recap:1. For part (I), using the symmetry condition and the discriminant condition, we found a = -1 and b = 2, so f(x) = -x² + 2x.2. For part (II), substituting f(x) into g(x), we got g(x) = (1/3)x³ + x² - 3x. Took the derivative, found critical points, evaluated at critical points and endpoints, and found that the maximum is 9 at x = 3.Everything seems to check out.**Final Answer**(I) The quadratic function is boxed{-x^2 + 2x}.(II) The maximum value of ( g(x) ) on ([0, 3]) is boxed{9}.