Use derivatives to sketch the graph of a function

Let

$f(x) = x + \cos x.$

Find all points such that $f'(x) = 0$ ;
Determine the intervals on which $f$ is monotonic by examining the sign of $f'$ ;
Determine the intervals on which $f'$ is monotonic by examining the sign of $f''$ ;
Sketch the graph of $f$ .

We take the derivative,
$f'(x) = 1 - \sin x.$

Thus,

$f'(x) = 0 \quad \implies \quad 1 - \sin x = 0 \quad \implies \quad x = \left(2n + \frac{1}{2}\right) \pi.$
$f$ is increasing for all $x$ since $1 - \sin x \geq 0$ for all $x$ .
Taking the second derivative,
$f''(x) = - \cos x.$

Thus, $f'$ is increasing if $\left( 2n + \frac{1}{2} \right) \pi < x < \left( 2n + \frac{3}{2} \right) \pi$ and is decreasing if $\left( 2n - \frac{1}{2} \right) \pi < x < \left( 2n + \frac{1}{2} \right) \pi$ .
We sketch the curve,