Question

If the area enclosed between $f(x)=\min .\left(\cos ^{-1}(\cos x)\right.$, $\left.\cot ^{-1}(\cot x)\right)$ and $x$-axis in $x \in(\pi, 2 \pi)$ is $\frac{\pi^{2}}{k}$ where $k \in N$, then $k$ is equal to

Answer 1

Given, $f(x)\begin{cases}x-\pi & ; \pi < x \leq \frac{3 \pi}{2} \\ 2 \pi-x & ; \frac{3 \pi}{2} < x < 2 \pi\end{cases}$

Clearly, required area
$=$ area of shaded portion of $\triangle A B C=\frac{1}{2} \times \frac{\pi^{2}}{2}$
$=\frac{\pi^{2}}{4}=\frac{\pi^{2}}{k}$
$\therefore$ On comparing, we get $k=4$.