# How do you find the critical values for f(x)=x^(2/3)+x^(-1/3)?

Oct 26, 2015

Find values in the domain of $f$ at which $f ' \left(x\right)$ does not exist or $f ' \left(x\right) = 0$ . See the explanation.

#### Explanation:

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

$\text{Dom} \left(f\right) = \mathbb{R} - \left\{0\right\}$

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

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

$= \frac{2 x - 1}{3 {x}^{\frac{4}{3}}}$

$f ' \left(x\right)$ does not exist at $x = 0$, which is not in $\text{Dom} \left(f\right)$.

$f ' \left(x\right) = 0$ at $x = \frac{1}{2}$ which is in $\text{Dom} \left(f\right)$.

The only critical number for $f$ is $\frac{1}{2}$.