# How do you find the inflections points for g(x) = 3x^4 − 6x^3 + 4?

May 6, 2015

Inflection points are points on the graph at which the concavity changes. To find them, we need to investigate concavity, and to do that we need the sign of the second derivative.

$g \left(x\right) = 3 {x}^{4} - 6 {x}^{3} + 4$

$g ' \left(x\right) = 12 {x}^{3} - 18 {x}^{2}$

$g ' ' \left(x\right) = 36 {x}^{2} - 36 x = 36 x \left(x - 1\right)$

Obviously, $g ' ' \left(x\right)$ is never undefined and is $0$ at $x = 0 , 1$

Investigating the sign of $g ' ' \left(x\right)$, we see that:

On $\left(- \infty , 0\right)$, $36 x$ is negative and $x - 1$ is negative, so $g ' ' \left(x\right)$ is positive.

On $\left(0 , 1\right)$, $36 x$ is positive and $x - 1$ is negative, so $g ' ' \left(x\right)$ is negative.

On $\left(1 , \infty\right)$, $36 x$ is positive and $x - 1$ is positive, so $g ' ' \left(x\right)$ is positive.

The sign of $g ' ' \left(x\right)$ (and hence the concavity) changes at $x = 0$ and at $x = 1$.

Inflection points are points on the graph at which the concavity changes, so we'll need the $y$ values as well.

The inflection points are: $\left(0 , 4\right)$ and $\left(1 , 1\right)$.