Question

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

Answer 1

The point `(0,0)`.

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'(x) = cos^2(x) + x(-sin(2x) + 2sin(x) + xcos(x))`

`f''(x) = -2sin(2x) + 2sin(x) + x(-2cos(2x) + 4cos(x) - xsin(x))`

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''(pi/2) > 0` and `f''(-pi/2) < 0`