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a) Xét $f(x)=\sin x-x+\frac{x^3}{6}, \forall x \in R$
$f'(x)=\cos x-1+-2\sin^2 \frac{x}{2}+\frac{x^2}{2}$
$=\frac{1}{2}[\frac{x^2}{2}-(\sin \frac{x}{2})^2]>0, \forall x >0$ ( Vì $|\sin t|<t, \forall x >0$)
$\Rightarrow f(x)>f(0)=0, \forall x >0$ $\Rightarrow \sin x>x-\frac{x^3}{6}, \forall x >0$
b) Xét $f(x)=\sin x+\tan x-2x, x\in (0;\frac{\pi}{2})$
$ f'(x)=\cos x+\frac{1}{\cos^2 x}-2>\cos^2 x+\frac{1}{\cos^2 x}-2$
$ \geq (\cos x-\frac{1}{\cos x})^2\geq 0, \forall x\in (0;\frac{\pi}{2}) $
$\Rightarrow f(x)>f(0)=0, \forall x\in (0;\frac{\pi}{2}) \Rightarrow \sin x+\tan x>2x, \forall x\in (0;\frac{\pi}{2})$
c) Xét $f(x)=2\sin x+\tan x-3x, \forall x\in (0;\frac{\pi}{2})$
$f'(x)=2\cos x+\frac{1}{\cos^2 x}-3=\cos x+\cos x+\frac{1}{\cos^2 x}-3$
$>3\sqrt[3]{\cos x.\cos x.\frac{1}{\cos^2 x}}-3=0, \forall x\in (0;\frac{\pi}{2})$
$\Rightarrow f(x)>f(0)=0, \forall x\in (0;\frac{\pi}{2})$
$\Rightarrow 2\sin x+\tan x>3x, \forall x\in (0;\frac{\pi}{2})$
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