Bài 105439

Question

Chứng minh rằng:
$-(1+x^{2})^{n}\leq (1-x^{2})^{n}+(2x)^{n}\leq (1+x^{2})^{n},\forall x \in R,\forall n\in N$\$\left\{ \begin{array}{l}0,1 \end{array} \right.\left. \right \}$

score 0 · Answer 1

BĐT cần chứng minh:
$\Leftrightarrow |(\frac{1-x^{2}}{1+x^{2}})^{n}+(\frac{2x}{1+x^{2}})^{n}|\leq 1 (1)$
Đặt:$x=\tan \frac{\alpha}{2},\alpha \in (-\pi,\pi)$
$\Rightarrow \begin{cases}\frac{1-x^{2}}{1+x^{2}}=\frac{1-\tan^{2} \frac{\alpha}{2}}{1+\tan^{2} \frac{\alpha}{2}}=cos^2\frac{\alpha}{2}-sin^2\frac{\alpha}{2}=\cos \alpha \\ \frac{2x}{1+x^{2}}= \frac{2\tan \frac{\alpha}{2}}{1+\tan^{2} \frac{\alpha}{2}}=2tan\frac{\alpha}{2}cos^2\frac{\alpha}{2}=\sin \alpha \end{cases}$
$(1)\Leftrightarrow |\sin ^{n}\alpha+\cos ^{n} \alpha|\leq 1 (2)$
Ta có:
$|\sin ^{n}\alpha+\cos ^{n} \alpha|\leq |\sin\alpha|^{n}+|\cos\alpha|^{n}$$\leq |\sin\alpha|^{2}+|\cos\alpha|^{2}=1$
( vì $n\geq 2, 0\leq |sin\alpha|, |cos\alpha|\leq 1$)
$\Rightarrow (2)$ đúng.
$\Rightarrow $(ĐPCM)

Bài 105439

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Bài 105439

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