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

Answer

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Answer

Solve for $C$ in the equation $f'(C) = \frac{f(4) - f(1)}{4 - 1}$: $C = \boxed{\sqrt{7}}$

Steps

Step 1 ：Find the derivative of the function: $f'(x) = 3x^2 - 8$

Step 2 ：Solve for $C$ in the equation $f'(C) = \frac{f(4) - f(1)}{4 - 1}$: $C = \boxed{\sqrt{7}}$