Problem

Find $f$ such that $f^{\prime}(x)=10 x-7, f(9)=0$

Answer

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Answer

\(\boxed{f(x) = 5x^2 - 7x - 342}\) is the final answer.

Steps

Step 1 ：Given the derivative of the function \(f'(x) = 10x - 7\) and a point on the function \((9,0)\).

Step 2 ：Integrate the derivative to find the function \(f(x)\). The integral of \(10x - 7\) is \(5x^2 - 7x + C\), where \(C\) is the constant of integration.

Step 3 ：Substitute the point \((9,0)\) into the function to solve for \(C\). So, \(0 = 5*9^2 - 7*9 + C\). Solving this equation gives \(C = -342\).

Step 4 ：Substitute \(C = -342\) back into the function to get the final function \(f(x) = 5x^2 - 7x - 342\).

Step 5 ：\(\boxed{f(x) = 5x^2 - 7x - 342}\) is the final answer.

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