Question

Find the intervals of monotonicity of the following functions.
$f\left(x\right)=x-\cos x$

Answer 1

$f\left(x\right)=x-\cos x$

$f^{{}^{\prime }}\left(x\right)=1+\sin x$

$-1\le \sin x\le 1$

$0\le 1+\sin x\le 2$

$f^{{}^{\prime }}\left(x\right)\ge 0$

$\therefore f\left(x\right)$ will never be monotonically decreasing.

For $f\left(x\right)$ to be monotonically increasing, $\sin x\ne -1$

$x\in R-\frac{(4n+3)\pi }{2},n\in I$