# How do you find points of inflection of the function function h(x) = −x^4 + x^2 − 1?

Apr 21, 2015

The points of inflection are ${x}_{1} = - \frac{\sqrt{6}}{6}$ ${x}_{2} = \frac{\sqrt{6}}{6}$

To calculate points of inflection you have to find zeros of the 2nd derivative and check if it changes sign at those points.

$h ' \left(x\right) = - 4 {x}^{3} + 2 x$
$h ' ' \left(x\right) = - 12 {x}^{2} + 2$

Looking for the points where $h ' ' \left(x\right) = 0$

$- 12 {x}^{2} + 2 = 0 / / : 2$
$- 6 {x}^{2} + 1 = 0$
$6 {x}^{2} = 1$
${x}^{2} = \frac{1}{6}$
${x}_{1} = - \frac{\sqrt{6}}{6}$ or ${x}_{2} = \frac{\sqrt{6}}{6}$

Checking if the derivative changes sign

graph{-12x^2+2 [-4.506, 4.505, -2.252, 2.254]}

From the graph we see that the derivative changes sign in both points ${x}_{1}$ and ${x}_{2}$ so they are the points of inflection.