Given f^{prime}(x)=-2 x+8, compute f(6)-f(3)

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Step:1 ：We are given the derivative of the function, (f&#x27;(x) = -2x + 8).Step:2 ：To find the original function, we need to integrate the derivative. The integral of (-2x + 8) is (-x^2 + 8x + C), where (C) is the constant of integration.Step:3 ：We are asked to find (f(6) - f(3)), which is ((-6^2 + 8*6 + C) - (-3^2 + 8*3 + C)).Step:4 ：Notice that the (C) terms cancel out, so we are left with ((-6^2 + 8*6) - (-3^2 + 8*3)).Step:5 ：Simplifying this expression gives us (-3).Step:6 ：So, (f(6) - f(3) = -3).Step:7 ：Final Answer: (boxed{-3})

Answer

Step: 1 ：We are given the derivative of the function, (f&#x27;(x) = -2x + 8).Step: 2 ：To find the original function, we need to integrate the derivative. The integral of (-2x + 8) is (-x^2 + 8x + C), where (C) is the constant of integration.Step: 3 ：We are asked to find (f(6) - f(3)), which is ((-6^2 + 8*6 + C) - (-3^2 + 8*3 + C)).Step: 4 ：Notice that the (C) terms cancel out, so we are left with ((-6^2 + 8*6) - (-3^2 + 8*3)).Step: 5 ：Simplifying this expression gives us (-3).Step: 6 ：So, (f(6) - f(3) = -3).Step: 7 ：Final Answer: (boxed{-3})

Given $f^{\prime}(x)=-2 x+8$, compute $f(6)-f(3)$

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