Question

Let $f:R\to R$ satisfy relation $f\left(x\right)f\left(y\right)-f\left(xy\right)=x+y\forall x,y\in R$ and $f\left(1\right)>0$. If $h\left(x\right)=f\left(x\right)f^{-1}\left(x\right)$, then length of longest interval in which $h(\sin x+\cos x)$ is strictly decreasing is

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (B)

$\frac{\pi }{2}$

$\begin{array}{c}h\left(x\right)=f\left(x\right)f^{-1}\left(x\right)\\ f\left(x\right)=\sin x+\cos x\\ =\sqrt{2}\left(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x\right)\\ =\sqrt{2}\left(\sin \left(x+\frac{\pi }{4}\right)\right)\\ f^{-1}\left(x\right)={\sin }^{-1}\left(\frac{x}{\sqrt{2}}-\frac{\pi }{4}\right)\\ f\left(x\right)f^{-1}\left(x\right)=\sqrt{2}\left(\sin \left(x+\frac{\pi }{4}\right)\right){\sin }^{-1}\left(\frac{x}{\sqrt{2}}-\frac{\pi }{4}\right)\end{array}$
Longest increasing interval $\frac{\pi }{2}$