# What are the critical values, if any, of  f(x)= x cos x −3 sin x +2x in [0,2pi]?

Nov 19, 2017

$0$, $2 \pi$ and a value near $4$ Please see below.

#### Explanation:

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

By inspection $x = 0$ and $x = 2 \pi$ are solutions.

$f ' \left(\pi\right) > 0$ and $f ' \left(\frac{3 \pi}{2}\right) < 0$ so there is another critical number in $\left[\pi , \frac{3 \pi}{2}\right]$.

Numerical or technological techniques find the solution at about $4$.