Let M and m respectively be the maximum and minimum values of the function f(x)=^-1( x+ x)…

Question

Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)=	an^{-1}(\sin x+\cos x)$ in $\left[0,\dfrac{\pi}{2}ight]$, then the value of $	an(M-m)$ is

Accepted Answer

For $x\in\left[0,\dfrac{\pi}{2}\right]$, $\sin x+\cos x\in[1,\sqrt2]$ (since $-\sqrt2\le A\sin x+B\cos x\le\sqrt2$ with $A=B=1$, restricted to this interval it attains $1$ and $\sqrt2$).

So $m=\tan^{-1}(1)=\dfrac{\pi}{4}$, $M=\tan^{-1}(\sqrt2)$.

$$\tan(M-m)=\tan\left(\tan^{-1}\sqrt2-\tan^{-1}1\right)=\frac{\sqrt2-1}{1+\sqrt2}=(\sqrt2-1)^2=3-2\sqrt2.$$

Solutions · 1