Let f x = 2 cos - 1 x + 4 cot - 1 x - 3 x 2 - 2 x + 10 , x ∈ - 1 , 1 . If a , b is the range of the…

Given $f\left(x\right)=2{\cos }^{-1}x+4{\cot }^{-1}x-3x^2-2x+10$

$f^{\text{'}}\left(x\right)=\frac{-2}{\sqrt{1-x^2}}-\frac{4}{1+x^2}-6x-2$

$=-2\left[\frac{1}{\sqrt{1-x^2}}+\frac{2}{1+x^2}+3x+1\right]$

$f^{\text{'}}\left(x\right)<0 \Rightarrow f\left(x\right)$ is a decreasing function

Now $f\left(1\right)=\pi +5$ and $f\left(-1\right)=5\pi +5$

So, range: $\left[a,b\right]\equiv \left[\pi +5,5\pi +5\right]$

$a=\pi +5, b=5\pi +5\Rightarrow 4a-b=11-\pi$