<s>[INST] Proof of $\sin2x+x\sin^2x \lt\dfrac{1}{4}x^2+2$ How can be proven the following inequality?
$$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
Thanks
 [/INST] $$\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$
$$\sin(x)(2\cos(x)+x\sin(x))\lt\dfrac{1}{4}x^2+2$$
while the right side is positive for all $x \in \mathbb{R}$ it is enoght that we show:
$$|\sin(x)|\cdot|(2\cos(x)+x\sin(x))|\lt\dfrac{1}{4}x^2+2$$
by the Cauchy–Schwarz inequality we have:
$$|\sin(x)|\cdot\sqrt{|4+x^2|.|\sin(x)^2+\cos(x)^2|}\lt\dfrac{x^2+4}{4}+1$$
$$|\sin(x)|\lt\dfrac{\sqrt{|4+x^2|}}{4}+\dfrac{1}{\sqrt{|4+x^2|}}$$
and the second side is greater than 1 by arithmetic and geometric inequality and equality.
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