What are the critical points of $f(x) = x^(1/3)*(x+8)$ ?

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Answer 1 · 2016-04-08T23:43:22

The critical numbers are $0$ and $-2$ .

A critical number for $f$ is a number, c, in the domain of $f$ such that $f'C9)=0$ or #f'(c) does not exist.

For $f(x) = x^(1/3)(x-8)$ , the domain is $RR$ .

And differentiating, we get

$f'(x) = 1/3x^(-2/3)(x-8) + x^(1/3) (1)$

$= (x-8)/(3x^(2/3)) +x^(1/3)$

$= (x-8)/(3x^(2/3)) +(3x)/(3x^(2/3))$

$= (4x+8)/(3x^(2/3)$ .

So, $f'(x) = 0$ at $x = -2$ and $f'(x)$ does not exists at $x=0$ .

Both are in the domain of $f$ , so both are critical numbers for $f$ .

What are the critical points of f(x) = x^(1/3)*(x+8) f(x) = x^(1/3)*(x+8)?