Question

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

Answer 1

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

Explanation:

`f'(x) = xsinx-2cosx+2`

By inspection `x=0` and `x= 2pi` are solutions.

`f'(pi) > 0` and `f'((3pi)/2) < 0` so there is another critical number in `[pi,(3pi)/2]`.

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