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

Sep 5, 2016

See below.

#### Explanation:

Rewrite: $f \left(x\right) = {x}^{\frac{4}{3}} + 4 {x}^{\frac{1}{3}}$

The domain of $f$ is all real numbers.

Differentiate:

$f ' \left(x\right) = \frac{4}{3} {x}^{\frac{1}{3}} + \frac{4}{3} {x}^{- \frac{2}{3}}$

$= \frac{4 \left(x + 1\right)}{3 {\sqrt[3]{x}}^{2}}$

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