If \(f(x)=16\left(\left(\sec ^{-1} x\right)^2+\left(\operatorname{cosec}^{-1} x\right)^2\right)\) then the sum of max. a…

\(f(x)=16\left({\left({\sec }^{-1}x\right)}^{2}+{\left({\csc }^{-1}x\right)}^{2}\right)\\ \Rightarrow f(x)=16\left({\left({\sec }^{-1}x\right)}^{2}+{\left(\frac{\pi }{2}-{\sec }^{-1}x\right)}^{2}\right)\\ \Rightarrow f(x)=32\left({\left({\sec }^{-1}x-\frac{\pi }{4}\right)}^{2}+\frac{{\pi }^{2}}{16}\right)\\ \Rightarrow f{(x)}_{min}=32\times \frac{{\pi }^{2}}{16}=2{\pi }^{2}\\ \Rightarrow \mathrm{f}{(\mathrm{x})}_{\max }=32\left({\left(\pi -\frac{\pi }{4}\right)}^{2}+\frac{{\pi }^{2}}{16}\right)=20{\pi }^{2}\\ \Rightarrow \mathrm{f}{(\mathrm{x})}_{\min }+\mathrm{f}{(\mathrm{x})}_{\max }=22{\pi }^{2}\)