How do you find the critical numbers for $f(x) = x^(1/3) (x+4)$ to determine the maximum and minimum?

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Answer 1 · 2016-09-05T17:04:56

See below.

Rewrite: $f(x) = x^(4/3)+4x^(1/3)$

The domain of $f$ is all real numbers.

Differentiate:

$f'(x) = 4/3 x^(1/3)+4/3 x^(-2/3)$

$= (4(x+1))/(3 root(3)x^2)$

$f'(x)$ does not exists at $0$ and $f'(x) = 0$ at $-1$ . Both are in the domain of $f$ , so both are critical numbers.

How do you find the critical numbers for f(x) = x^(1/3) (x+4) f(x) = x^(1/3) (x+4) to determine the maximum and minimum?