Min value of a trigonometric expression

Question

What is the value of $\sin(x)$ for the maximum value of $(5+3\sin(x))^2 (7-3\sin(x))^3$.

Answer 1

HINT

Let $z = \sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.

UPDATE

As mentioned in the comments below, you are only optimizing over $-1 \le z \le 1$.

Answer 2

Hint:

AM GM inequality

$$\dfrac{2\cdot3(5+3\sin x)+3\cdot2(7-3\sin x)}{2+3}\ge\sqrt[5]{3^2(5+3\sin x)^22^3(7-3\sin x)^3}$$

The equality occurs if $3(5+3\sin x)=2(7-3\sin x)$

Answer 3

AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...

let $z=\sin(x)$, $f(z)=g(z)^2h(z)^3$

$$ f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh') $$

since $-1 \le z \le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.