Mathematics - Application of Derivative Question with Solution | TestHub

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