Given the function $f (x) = x (x^{2} - 8) - 5$ 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 $f^{'} (C) = \frac{f (4) - f (1)}{4 - 1}$ : $C = \sqrt{7}$

Steps

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

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