Problem
Given the function $f(x)=x\left(x^{2}-8\right)-5$ satisfies the hypothesis of the Mean Value Theorem on the interval $[1,4]$, find a number $C$ in the interval $(1,4)$ which satisfies this theorem. $\sqrt{5}$ 12 $\sqrt{7}$ 5
Solution
Step 1 ：Find the derivative of the function: $f'(x) = 3x^2 - 8$
Step 2 ：Solve for $C$ in the equation $f'(C) = \frac{f(4) - f(1)}{4 - 1}$: $C = \boxed{\sqrt{7}}$
