How to prove the following inequality:
$\sin{x}+x\cos{x}\le\sqrt{1+x^{2}}$ for all $x\in\mathbb{R}$.
It seems to be related the function $f(x)=x\sin{x}$ and M.V.T., but, I'm not sure about that.
2026-04-06 05:48:03.1775454483
Inequality for trigonometric function
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Use the Cauchy Schwarz inequality
$$(\sin(x)+x\cos(x))^2\le (1^2+x^2)(\sin^2(x)+\cos^2(x))$$