[tex]f(x)=\sin x-\cos x=-\sqrt{2}\sin \left ( \frac{\pi}{4}-x \right )\Rightarrow f'(x)=\sqrt{2}\sin \left ( x+\frac{\pi}{4} \right )\\f'(x)=0\Rightarrow \sqrt{2}\sin \left ( x+\frac{\pi}{4} \right )=0\Leftrightarrow \sin \left ( x+\frac{\pi}{4} \right )=0\Rightarrow x=-\frac{\pi}{4}+\pi k,k \in \mathbb{Z}\\x\in [0,\pi]\Rightarrow k=1\Rightarrow x=\frac{3\pi}{4}\\x=0\Rightarrow f(0)=-1\\[/tex][tex]x=\pi \Rightarrow f(\pi)=1\\x=\frac{3\pi}{4}\Rightarrow f\left ( \frac{3\pi}{4} \right )=\sqrt{2}\\ \min f(x)=-1, \ \ \max f(x)=\sqrt{2}[/tex]