Find the maximum value of + . - Vidyadip

Question

Find the maximum value of $\sin \theta+\cos \theta$.

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Answer

Given $ f(\theta)=\sin \theta+\cos \theta, \theta \in[0, \pi]$
$\therefore f^{\prime}(\theta)=\cos \theta-\sin \theta$
$f^{\prime}(\theta)=0$
$\Rightarrow \sin \theta-\cos \theta=0 $
$\Rightarrow \tan \theta=1$
$\therefore \theta=\frac{\pi}{4}$
hence $ f(0)=\sin 0+\cos 0=1$
$f\left(\frac{\pi}{4}\right)=\sin \frac{\pi}{4}+\cos \frac{\pi}{4}$
$=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}$
$f(\pi)=\sin \pi+\cos \pi=0-1=-1$
$\therefore \text { Absolute maximum value }=\sqrt{2}$

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