Question

Find all values of $x$ for which the function $y=\sin x{\cos }^{2}x-1$ assumes the least value. What is that value?

Answer 1

$y=\sin x{\cos }^{2}x-1$

$\frac{dy}{dx}={\cos }^{3}x-2{\sin }^{2}x\cos x=0$

$\cos x=0\Rightarrow x=(2n+1)\frac{\pi }{2}$

$\cot x=\pm \sqrt{2}\Rightarrow x=n\pi +{\cot }^{-1}\left(\sqrt{2}\right)$

At $x=(2n+1)\frac{\pi }{2},n\pi +{\cot }^{-1}\left(\sqrt{2}\right)$ the value of y is minimum.