The absolute maximum value of the function $f(x)=sinx+cosx, x \in [0, \pi]$ is :

$\sqrt{2}$ 2 1 $\frac{1}{\sqrt{2}}$

$\sqrt{2}$

The correct answer is Option (1) → $\sqrt{2}$ $f(x)=\left(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x\right)^{\sqrt{2}}$ $=\sqrt{2}(\sin x\cos(π/4)+\cos x\sin(π/4))$ $=\sqrt{2}\sin(x+π/4)$    $x∈[0,π]$ at $x=π/4$ max. exists $f(π/4)=f_{max}=\sqrt{2}$