Question

Find maximum and minimum value of $\sin x\sin ({60}^{\circ }-x)sin({60}^{\circ }+x)$.

Answer 1

Let $f\left(x\right)=\sin x\sin ({60}^{\circ }-x)\sin ({60}^{\circ }+x)=\sin x({\sin }^2{60}^{\circ }-{\sin }^{2}x)=\frac{1}{4}(3\sin x-4{\sin }^{3}x)=\frac{1}{4}\sin 3x$

As $\sin 3x\in [-1,1]⟹\frac{1}{4}\sin 3x\in [-\frac{1}{4},\frac{1}{4}]⟹f\left(x\right)\in [-\frac{1}{4},\frac{1}{4}]$

So the minimum and maximum value are $-\frac{1}{4},\frac{1}{4}$ respectively