TITLE: Extreme values of $2\cos^2 \frac{x}{2}\cdot\sin^2 x$ without calculus
QUESTION [0 upvotes]: To find the extreme values of
$$2\cos^2 \frac{x}{2}\cdot\sin^2 x$$

I Have not learnt calculus yet but this is a question within our syllabus .which made me to think if this can be done with it .
I am not able approach this problem any help is appreciated.

REPLY [0 votes]: Writing $x=2y$
$$\cos y\sin2y=2(1-\sin^2y)\sin y$$
$$\dfrac{2\sin^2y+1-\sin^2y+1-\sin^2y}3\ge\sqrt[3]{2\sin^2y(1-\sin^2y)^2}$$
$$\implies\left(\sqrt2\sin y(1-\sin^2y)\right)^{2/3}\le\dfrac23$$
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