# What are the points of inflection, if any, of f(x) = -3x^4-2 x^3 +9 ?

In order to find the points of inflection we need to find the roots of

$f ' ' \left(x\right)$ eg the second derivative of the function.

Hence from

$f ' ' \left(x\right) = 0 \implies - 36 {x}^{2} - 12 x = 0 \implies 12 x \cdot \left(3 x + 1\right) = 0$

so the roots are

$x = 0$ and $x = - \frac{1}{3}$

The possible points of inflection are $x = 0$ and $x = - \frac{1}{3}$ .

But Inflection points are where the function changes concavity.This is happening only for $x = 0$

So the inflection point is $\left(0 , f \left(0\right)\right)$ or $\left(0 , 9\right)$