Question

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

• A. $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
• B. $\left(-\frac{\pi}{2}, \frac{\pi}{4}\right)$
• C. $\left(0, \frac{\pi}{2}\right)$
• D. $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Answer 1

Answer: (B)

$f'(x) = \frac{1}{1+(sin\,x + cos\,x)^2} (cos\,x - sin\,x)$
$f(x)$ is increasing if
$cos\,x - sin\,x > 0$
or $cos\,x > sin\,x$
Hence, $f(x)$ is increasing when
$x\in\left(-\frac{\pi}{2}, \frac{\pi}{4}\right)$