Question

If the range of functions $f\left(x\right)={\left({\cos }^{-1}\frac{x}{3}\right)}^2+\pi {\sin }^{-1}\frac{x}{3}-{\left({\sin }^{-1}\frac{x}{3}\right)}^2+\frac{{\pi }^2}{4}(x^2+2x+1)$ is $[a,b]$. then the value of $\frac{b}{a}$ is

Answer 1

Let ${\sin }^{-1}\left(\frac{x}{3}\right)=t$ and $g\left(x\right)={\left({\cos }^{-1}\frac{x}{3}\right)}^2+\pi {\sin }^{-1}\frac{x}{3}-{\left({\sin }^{-1}\frac{x}{3}\right)}^2$
$\Rightarrow g\left(x\right)={(\frac{\pi }{2}-t)}^2+\pi t-t^2\Rightarrow g\left(x\right)=\frac{{\pi }^2}{4}\Rightarrow f\left(x\right)=\frac{{\pi }^2}{4}\left(\right(x+1{)}^2+1)$

Now
$-1\le \frac{x}{3}\le 1\Rightarrow -3\le x\le 3\Rightarrow -2\le x+1\le 4\Rightarrow 0\le (x+1{)}^2\le 16\Rightarrow 1\le (x+1{)}^2+1\le 17\Rightarrow \frac{{\pi }^2}{4}\le \frac{{\pi }^2}{4}\left(\right(x+1{)}^2+1)\le 17\frac{{\pi }^2}{4}$