Problem
For the function $f(x)=8 x^{2}-3 x+5$, find $f^{\prime \prime}(x)$. Then find $f^{\prime \prime}(0)$ and $f^{\prime \prime}(5)$. $f^{\prime \prime}(x)=\square($ Simplify your answer.) Select the correct choice below and fill in any answer boxes in your choice. A. $f^{\prime \prime}(0)=\square$ (Simplify your answer.) B. $f^{\prime \prime}(0)$ is undefined. Select the correct choice below and fill in any answer boxes in your choice. A. $f^{\prime \prime}(5)=\square$ (Simplify your answer) B. $f^{\prime \prime}(5)$ is undefined.
Solution
Step 1 ：Find the first derivative of the function, \(f'(x)\). The derivative of \(x^{2}\) is \(2x\), the derivative of \(x\) is \(1\), and the derivative of a constant is \(0\). So, \(f'(x) = 2*8x - 3 + 0 = 16x - 3\).
Step 2 ：Find the second derivative of the function, \(f''(x)\). The derivative of \(x\) is \(1\), and the derivative of a constant is \(0\). So, \(f''(x) = 16 - 0 = 16\).
Step 3 ：Find \(f''(0)\) and \(f''(5)\). Since \(f''(x)\) is a constant, \(f''(0) = f''(5) = 16\).
Step 4 ：So, the answers are: \(\boxed{f''(0) = 16}\) and \(\boxed{f''(5) = 16}\).
