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  4. If f(x)=(4/π) cot -1 x-(π/4 cot -1(-x))+2, then the maximum admissible value of f(x) will be

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Q. If $f(x)=\frac{4}{\pi} \cot ^{-1} x-\frac{\pi}{4 \cot ^{-1}(-x)}+2$, then the maximum admissible value of $f(x)$ will be

Inverse Trigonometric Functions

Solution:

$f(x)=2+\frac{4}{\pi}\left(\pi-\cot ^{-1}(-x)\right)-\frac{\pi}{4 \cot ^{-1}(-x)} \left[\Theta \cot ^{-1}(x)>0\right]$
$=2+4-\left[\frac{4}{\pi} \cot ^{-1}(-x)+\frac{\pi}{4 \cot ^{-1}(-x)}\right] $
$\leq 2+4-2=4$
$f ( x )_{\max }=4 $