# How do you find the inflection points of the graph of the function:  f(x)=x^(1/3)?

Aug 31, 2015

Examine the behaviour of $f ' ' \left(x\right) = - \frac{2}{9} {x}^{- \frac{5}{3}}$ to find the inflexion point at $x = 0$

#### Explanation:

Given $f \left(x\right) = {x}^{\frac{1}{3}}$

$f ' \left(x\right) = \left(\frac{1}{3}\right) {x}^{- \frac{2}{3}}$ which is undefined at $x = 0$

$f ' ' \left(x\right) = - \frac{2}{9} {x}^{- \frac{5}{3}}$ which is also undefined at $x = 0$

Note that, where it is defined $f ' ' \left(x\right)$ is always non-zero.

There are therefore no inflexion points in the domain of $f ' ' \left(x\right)$, that is in $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$.

However, $x = 0$ is an inflexion point of $f \left(x\right)$, since for any $\epsilon > 0$ we find:

$f ' ' \left(\epsilon\right) < 0$ and $f ' ' \left(- \epsilon\right) > 0$

So the sign of $f ' ' \left(x\right)$ changes at $x = 0$