# How do you find the critical numbers of  f(x)=x^(1/5)-x^(-4/5)?

Jul 3, 2017

The only critical number is $x = - 4$

#### Explanation:

The critical numbers of a function are the points where its derivative equals zero, which means they are the solutions of the equation:

$\frac{\mathrm{df}}{\mathrm{dx}} = 0$

$\frac{1}{5} {x}^{- \frac{4}{5}} + \frac{4}{5} {x}^{- \frac{9}{5}} = 0$

$\frac{1}{5} {x}^{- \frac{4}{5}} \left(1 + \frac{4}{x}\right) = 0$

As ${x}^{- \frac{4}{5}} \ne 0$ for any value of $x$ the only solutions are given by:

$1 + \frac{4}{x} = 0$

$\frac{4}{x} = - 1$

$x = - 4$