Question

determine all critical points for the following function.

$f(x)=2x^{2}+\frac{32}{x}$

what is the derivative of $f(x)=2x^{2}+\frac{32}{x}$?

$f(x)=4x - \frac{32}{x^{2}}$

find the critical points, if any, of $f$ on the domain. select the correct choice below and, if necessary, fill in

a. the critical point(s) occur(s) at $x = \square$

(use a comma to separate answers as needed.)

b.

there are no critical points for $f(x)=2x^{2}+\frac{32}{x}$ on the domain.

Explanation:

Step1: Set the derivative equal to zero

Set \( f'(x)=4x - \frac{32}{x^{2}}=0\).
Multiply through by \(x^{2}\) (since \(x
eq0\) as \(x = 0\) is not in the domain of \(f(x)\)) to get \(4x^{3}-32 = 0\).

Step2: Solve for \(x\)

First, rewrite \(4x^{3}-32 = 0\) as \(4x^{3}=32\).
Then divide both sides by \(4\): \(x^{3}=8\).
Take the cube - root of both sides: \(x = 2\).

Answer:

A. The critical point(s) occur(s) at \(x = 2\)