Question

Let $S$ be set of domain of the function $f ( x )=\sqrt{\frac{\pi}{2}-\tan ^{-1} \sqrt{- x ^2+5 x -6}}$. If $\lambda=\alpha+\frac{1}{\alpha}$ where $\alpha \in S$ and $\lambda$ is an integer then find the value of $\left(\lambda^2\right)$.

Answer 1

For domain of function,
$-x^2+5 x+6 \geq 0 \Rightarrow x^2-5 x+6 \leq 0 $
$\Rightarrow (x-2)(x-3) \leq 0 $
$\Rightarrow x \in[2,3]$
$\text { Now, } \lambda=\alpha+\frac{1}{\alpha} \Rightarrow \lambda \in\left[\frac{5}{2}, \frac{10}{3}\right]$
So, integral value of $\lambda$ is 3 .
Hence, $\lambda^2=9$.