Question

If $f(x)=x-2 \sin x, 0 \leq x \leq 2 \pi$ is increasing in the interval $[a \pi, b \pi]$, then find value of $a+b$.

Answer 1

$f'(x)=1-2 \cos x \geq 0$
$\Rightarrow \cos x \leq \frac{1}{2}$
$\therefore x \in\left[\frac{\pi}{3}, \frac{5 \pi}{3}\right]$
$\therefore a+b=\frac{1}{3}+\frac{5}{3}=2$