I know that I can find the maximum of this function by using derivatives but is there an other way of finding the maximum that does not involve derivatives? Maybe use a well-known inequality or identity?
$f(x)=\sin(2x)+2\sin(x)$
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The idea is to use $\sin^2 x + \cos^2x = 1$ to reduce to dealing with only 1 trigonometric function, and then proceed as a standard 1-variable inequality.
We wish to find the maximum of $f(x) = \sin 2x + 2 \sin x = 2 \sin x ( 1 + \cos x )$.
It is clear that we may assume $\sin x \geq 0, \cos x \geq 0$, to maximize this product.
Let's consider $$[f(x)]^2 = 4 \sin^2 x ( 1 + \cos x)^2 = 4 (1-\cos^2x ) ( 1 + \cos x)^2 = 4 ( 1 - \cos x ) ( 1 + \cos x)^3 $$
By AM-GM, applied to $ 3(1-\cos x) , (1 + \cos x), (1 + \cos x), (1 + \cos x)$, we get that
$ \sqrt[4]{3 ( 1 - \cos x ) ( 1 + \cos x)^3} \leq \frac{ 6}{4}$, or that $ ( 1 - \cos x ) ( 1 + \cos x)^3 \leq \frac{27}{16}$.
Hence, $ [f(x)]^2 \leq \frac{27}{4}$ so $f(x) \leq \frac{3 \sqrt{3} } {2} $.
It remains to verify that equality can occur, which it does at $3(1-\cos x) = 1 + \cos x$ of $\cos x = \frac{1}{2}$.
Calvin Lin
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You beat me to it! +1 – amWhy Jun 21 '13 at 23:49
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Very nice! Exactly what I was looking for. Thanks Calvin – EricAm Jun 21 '13 at 23:51
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@AdamYac The motivation is pretty clear, and can be routine, if you're familiar with 1 variable inequalities. E.g. try to find the max of $ ( 1 + \sin x) ( 1 + \cos x)$. – Calvin Lin Jun 21 '13 at 23:53
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Edit: Never mind, I carried out the steps by hand and I see why now. – EricAm Jun 22 '13 at 18:22
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As Calvin Lin has observed $\sin x,\cos x>0 \implies 0\le x\le \frac\pi2$ $\implies 0\le 2x\le \pi\implies0\le \pi-2x\le \pi$
As sine function is concave in $[0,\pi]$ $$\sin2x+2\sin x$$
$$=\sin(\pi-2x)+\sin x+\sin x\le 3\sin\frac{(\pi-2x+x+x)}3=3\sin\frac{\pi}3=\frac{3\sqrt3}2$$ the equality occurs when $x=x=\pi-2x$ i.e., when $x=\frac\pi3$
lab bhattacharjee
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@AdamYac, how about this method (http://www.artofproblemsolving.com/Wiki/index.php/Jensen's_Inequality) – lab bhattacharjee Jun 22 '13 at 09:16
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I've never heard about Jensen's Inequality yet but I will look into it. Thanks lab b. – EricAm Jun 22 '13 at 10:45
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@AdamYac, my pleasure. Hope I could make the method clear and I believe this does not involve calculus – lab bhattacharjee Jun 22 '13 at 13:43
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Doesn't showing that something is concave/convex use calculus? Of course, you can try and do it by definition, but i think showing $\lambda \sin x + (1-\lambda) \sin y \leq \sin ( \lambda x +(1-\lambda)y) $ directly would be tricky. – Calvin Lin Jun 23 '13 at 16:17
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@CalvinLin, How about this method (http://planetmath.org/concavityofsinefunction) (linked in the answer)? – lab bhattacharjee Jun 23 '13 at 16:25
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@CalvinLin, For concavity: Is it not enough to refer to the unit circle and conclude that the tangent is always above the circle on the interval $x\in [0,\pi]$ – EricAm Jun 27 '13 at 23:28