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

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Answer 1 · 2015-12-18T23:28:38

The point $(0,0)$ .

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 $f''(-pi/2) < 0$

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