Question

If $f(x)=16\left(\left(\cos ^{-1} x\right)^{2}+\left(\sin ^{-1} x\right)^{2}\right)$, then find sum of min. & max. value of $f(x)$

• B. $22 \pi^{2}$
• C. $24 \pi^{2}$
• D. $18 \pi^{2}$
• E. $31 \pi^{2}$

Answer 1

Answer: (B)

$22 \pi^{2}$

Answer 2

The function $f(x)$ can be simplified using the identity $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$. By substituting $\sin^{-1} x = \theta$, we can express $f(x)$ in terms of $\theta$ and find its minimum and maximum values. The minimum value occurs at $\theta = 0$ and the maximum at $\theta = \frac{\pi}{2}$. Evaluating these gives the sum of minimum and maximum values as $22\pi^2$.