# How do you find the critical points of f(x)=sinx+cosx?

$y = \sin x + \cos x$
Use the Trig Identity $\sin + \cos x = \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)$.
$y = \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)$
$y$ min when $\sin \left(x + \frac{\pi}{4}\right) = - 1 \Rightarrow x + \frac{\pi}{4} = \frac{3}{2} \pi \Rightarrow x = \frac{5}{4} \pi$.
$y$ max when $\sin \left(x + \frac{\pi}{4}\right) = 1 \Rightarrow x + \frac{\pi}{4} = \sin \frac{\pi}{2} \Rightarrow x = \frac{\pi}{4}$.
In the interval $\left(0 , 2 \pi\right)$ there are $2$ answers: $\frac{\pi}{4}$ and $\frac{5}{4} \pi$.
When $x = \frac{\pi}{4} \Rightarrow y = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}$ (Max)
When $x = \frac{5}{4} \pi \Rightarrow y = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = - \sqrt{2}$ (Min)