What is the greatest value of $\sin x + \cos x$ on $[0, \frac{\pi}{2}]$?

Explanation

1. 01

   $f'(x) = - \sin x + \cos x = 0$ inside the interval at $x = \frac{\pi}{4}$.

2. 02

   Values: $f(0) = 1$, $f(\frac{\pi}{2}) = 1$, $f(\frac{\pi}{4}) = \sqrt{2}$; the greatest value is $\sqrt{2}$.