Question

The function f(x) = tan${}^{-1}$ (sinx + cosx) is an increasing function in-

• A. $(\pi /4,\pi /2)$
• B. $(-\pi /2,\pi /4)$
• C. $(0,\pi /2)$
• D. $(-\pi /2,\pi /2)$

Answer 1

Answer: (B)

$(-\pi /2,\pi /4)$

$\because f\left(x\right)={\tan }^{-1}(\sin x+\cos x)$
$\therefore f^{\prime }\left(x\right)=\frac{\cos x-\sin x}{1+{(\sin x+\cos x)}^2}=\frac{\sqrt{2}\cos (x+\frac{\pi }{4})}{1+{(\sin x+\cos x)}^2}$
$f\left(x\right)$ is increasing if $-\frac{\pi }{2}<x+\frac{\pi }{4}<\frac{\pi }{2}\Rightarrow -\frac{3\pi }{4}<x<\frac{\pi }{4}$
Hence, $f\left(x\right)$ is increasing when $x\in (-\frac{\pi }{2},\frac{\pi }{4})$