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

Nov 1, 2015

$\left(x , y\right) = \left(0 , 0\right)$ is an inflection point.

#### Explanation:

$y = \frac{1}{3} {x}^{3}$

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 2 x$

At inflection point, second derivative must be zero, which occurs at $x = 0$.

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} {|}_{x = 0} = 2 \left(0\right) = 0$

Now, do the first derivative test.

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{2}$ changes from positive to zero to positive in a small neighborhood around $x = 0$. $x = 0$ is an inflection point.