What is the maximum value of the function $\sin x + \cos x$ ?

Let $f(x) = \sin x + \cos x,x \in R$
$\Rightarrow f'(x) = \cos x - \sin x$

For critical points, $f(x) = 0$
$\Rightarrow \cos x - \sin x = 0 \Rightarrow \tan x = 1 \Rightarrow x = \cfrac{\pi }{4},\cfrac{{5\pi }}{4}$

To find the maximum value of the function , we have to evaluate $f\left( {\cfrac{\pi }{4}} \right)$ and

$f\left( {\cfrac{{5\pi }}{4}} \right)$
$f\left( {\cfrac{\pi }{4}} \right) = \sin \cfrac{\pi }{4} + \cos \cfrac{\pi }{4} = \sqrt 2$

$f\left( {\cfrac{{5\pi }}{4}} \right) = \sin \cfrac{{5\pi }}{4} + \cos \cfrac{{5\pi }}{4} = \sin \left( {\pi + \cfrac{\pi }{4}} \right) + \cos \left( {\pi + \cfrac{\pi }{4}} \right)$
$= - \sin \cfrac{\pi }{4} - \cos \cfrac{\pi }{4} = - \sqrt 2$

Therefore the maximum value of the function is $\sqrt 2$ at $x = \cfrac{\pi }{4}$