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  4. Let f(x)=2 cos -1 x+4 cot -1 x-3 x2-2 x+10, x ∈[- 1,1]. If [ a , b ] is the range of the function then 4 a - b is equal to:

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Q. Let $f(x)=2 \cos ^{-1} x+4 \cot ^{-1} x-3 x^{2}-2 x+10, x \in[- 1,1]$. If $[ a , b ]$ is the range of the function then $4 a -$ $b$ is equal to:

JEE MainJEE Main 2022Inverse Trigonometric Functions

Solution:

$f^{\prime}(x)=\frac{-2}{\sqrt{1-x^{2}}}-\frac{4}{1+x^{2}}-6 x-2$
$=-2\left[\frac{1}{\sqrt{1-x^{2}}}+\frac{2}{1+x^{2}}+3 x+1\right]$
$f^{\prime}(x)<0 \Rightarrow f(x)$ is a dec. function
$f(1)=\pi+5$
$f(-1)=5 \pi+5$
Range : $[a,b] =[\pi+5,5 \pi+5]$
$a =\pi+5, b=5 \pi+5 \Rightarrow 4 a-b=11-\pi$