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

Dec 18, 2015

The point $\left(0 , 0\right)$.

#### Explanation:

In order to find the inflection points of $f$, you have to study the variations of $f '$, and to do that you need to derivate $f$ two times.

$f ' \left(x\right) = {\cos}^{2} \left(x\right) + x \left(- \sin \left(2 x\right) + 2 \sin \left(x\right) + x \cos \left(x\right)\right)$

$f ' ' \left(x\right) = - 2 \sin \left(2 x\right) + 2 \sin \left(x\right) + x \left(- 2 \cos \left(2 x\right) + 4 \cos \left(x\right) - x \sin \left(x\right)\right)$

The inflection points of $f$ are the points when $f ' '$ is zero and goes from positive to negative.

$x = 0$ seems to be such a point because $f ' ' \left(\frac{\pi}{2}\right) > 0$ and $f ' ' \left(- \frac{\pi}{2}\right) < 0$