# Prove that $3x-x^3<\frac2{\sin2x}$

Prove that $$3x-x^3<\frac2{\sin2x},\forall x\in\left(0,\frac\pi2\right)$$

I have tried by proving that $$3x-x^3<\frac9{5\pi}x+\frac32<\frac2{\sin2x},\forall x\in\left(0,\frac\pi2\right)$$ with Jensen's inequality, but I hoped this problem would have a simpler solution.

HINT:

$$\sin 2x\le1 \implies \frac{2}{\sin 2x}\ge 2$$

and

$$3x-x^3\le 2$$

• Thank you, I was foolish to see this.
– mja
Sep 2 '15 at 16:16
• You're welcome. My pleasure. And no you're not foolish. I have missed the tree in the forest many times. It is easy to do. You're fine! Sep 2 '15 at 16:17

The range of the function $f(x)=3x-x^3$ for $x\in(0,\pi/2)$ is $f(x)\in(0,2)$.

For the function $g(x)=\dfrac2{\sin2x}$ for $x\in(0,\pi/2)$ the range is clearly $g(x)\in(2,\infty)$.

The inequality is quite clear now.

• Just like mine ;-)) Sep 2 '15 at 16:15
• Yeah well the mobile site does not display answers in real-time. Sep 2 '15 at 16:16
• No worry. I've posted answers like that many times. It's good to see alignment with our answers! Sep 2 '15 at 16:17

Take $f(x)=(3x-x^3)\sin(2x)$ over $I=\left[0,\frac{\pi}{2}\right]$. It is a non-negative function since $\pi^2<12$. Since over the same interval we have $\sin(2x)\leq \frac{4}{\pi^2}(2x)(\pi-2x)$, it is enough to prove that:

$$\forall x\in I,\qquad x^2(3-x^2)(\pi-2x)\leq \frac{\pi^2}{4}. \tag{1}$$ By differentiation we may locate the absolute maximum of $g(x)=x^2(3-x^2)(\pi-2x)$ over $I$ around $x=0.88552583$. In such a point $g(x)$ is about $2.38141<2.4674<\frac{\pi^2}{4}$, so $(1)$ holds.

• It's much simpler. $2\csc (2x)\ge 2$ and $3x-x^3\le 2$. Sep 2 '15 at 16:14
• @Dr.MV: oh, nice. I thought it was possible to separate the two functions by putting a line in between (since they are a convex and a concave function), but I didn't try the most simple approach :D Sep 2 '15 at 16:34
• I began thinking about ways to do this and then it just struck me. It's very easy to miss the tree from the forest so they say. Sep 2 '15 at 16:39
• @Dr.MV "sometimes you can't see the wood for the trees" is the phrase that springs to mind Sep 2 '15 at 18:10
• @DavidQuinn This is the saying that I was paraphrasing. Sep 2 '15 at 19:17