Find all the values of $x$ for which the function $y = \sin x – \cos^2 x -1$ assumes the least value. What is that value?
At first I found the first derivative to be $y' = \cos x + 2 \sin x \cos x$.
Critical point $0$, $-π/6$ (principal)
$y'' = – \sin x + 2(\cos 2x)$
Then substitution of $x$ by critical points I found minima. But my answer is incorrect.
Correct minimum value is $-9/4$
Best Answer
$$f(x)=\sin^2x+\sin{x}-2=\left(\sin{x}+\frac{1}{2}\right)^2-2.25\geq-2.25.$$ The equality occurs for $\sin{x}=-\frac{1}{2},$ which says that $-2.25$ is a minimal value.