# What are the points of inflection of f(x)= x(x^2+1) ?

Sep 16, 2017

Start by taking the second derivative..

#### Explanation:

...which is easier if you multiply through:

$\frac{d}{\mathrm{dx}} \left({x}^{3} + x\right) = 3 {x}^{2} + 1$

$\frac{{d}^{2} x}{\mathrm{dx}} ^ 2 = 6 x$

...and your inflection point is where this quantity goes from positive to negative, or vice versa.

This happens at point x = 0.

On a graph, an inflection point is where the curve shifts over from a left turn to a right turn, or vice versa:

graph{x(x^2 + 1) [-10, 10, -5, 5]}