Let M and m
Question:

Let $\mathrm{M}$ and $\mathrm{m}$ respectively be the maximum and minimum values of the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ in $\left[0, \frac{\pi}{2}\right]$, Then the value of $\tan (\mathrm{M}-\mathrm{m})$ is equal to:

  1. $2+\sqrt{3}$

  2. $2-\sqrt{3}$

  3. $3+2 \sqrt{2}$

  4. $3-2 \sqrt{2}$


Correct Option: , 4

Solution:

Let $g(x)=\sin x+\cos x=\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)$

$\mathrm{g}(\mathrm{x}) \in[1, \sqrt{2}]$ for $\mathrm{x} \in[0, \pi / 2]$

$f(x)=\tan ^{-1}(\sin x+\cos x) \in\left[\frac{\pi}{4}, \tan ^{-1} \sqrt{2}\right]$

$\tan \left(\tan ^{-1} \sqrt{2}-\frac{\pi}{4}\right)=\frac{\sqrt{2}-1}{1+\sqrt{2}} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}=3-2 \sqrt{2}$

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