The function f ( x ) = ^ - 1 ( x + x ) is an increasing function … - Vidyadip

MCQ

The function $f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$ is an increasing function in

A
$\left( {0,\frac{\pi }{2}} \right)$
B
$\;\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$
C
$\;\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)$
✓
$\;\left( { - \frac{\pi }{2},\frac{\pi }{4}} \right)$

✓

Answer

Correct option: D.

$\;\left( { - \frac{\pi }{2},\frac{\pi }{4}} \right)$

d
$f^{\prime}(x)=\frac{1}{1+(\sin x+\cos x)^{2}} \cdot(\cos x-\sin x)$

$f^{\prime}(x)=\frac{\cos x-\sin x}{2+\sin 2 x}$

If $f^{\prime}(x)>0$ then $f(x)$ is increasing function

For $ - \frac{\pi }{2}\,sin\,x$

Hence $y=f(x)$ is increasing in $\left(-\frac{\pi}{2}, \frac{\pi}{4}\right)$

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