Question

Find the maximum and minimum values of $f\left(x\right)=4x+\cot x,x\in (0,\pi )$

Answer 1

$f\left(x\right)=4x+\cot x$

$⟹f^{\prime }\left(x\right)=4-cose{c}^{2}x=0$

$⟹cosecx=\pm 2$

$⟹x=\frac{\pi }{6},\frac{5\pi }{6}$

Now, $f\left(\frac{\pi }{6}\right)=\frac{2\pi }{3}+\sqrt{3}=f_{max}\left(x\right)$

And, $f\left(\frac{5\pi }{6}\right)=\frac{5\pi }{6}-\sqrt{3}=f_{min}\left(x\right)$