# Min value of a trigonometric expression [closed]

Hik Aubergine 10/16/2018. 3 answers, 261 views

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

lab bhattacharjee 10/16/2018.

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 $$3(5+3\sin x)=2(7-3\sin x)$$

karakfa 10/16/2018.

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 $$z$$.

gt6989b 10/16/2018.

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 $$-1 \le z \le 1$$.