1. Cho x,y,z >0 thỏa xyz= $\frac{1}{2}$ . CMR:
$\frac{xy}{x^2.(y+z)}$ +$\frac{xz}{y^2.(x+z)}$ +$\frac{xy}{z^2.(x+y)}$ $\geq$ xy+yz+xz
2. Cho a,b,c >0 thỏa ab+bc+ca=1. CMR:
a. $\sqrt[]{b^2+1}$ +b. $\sqrt[]{c^2+1}$ +c. $\sqrt[]{a^2+1}$ $\geq$ 2
3. Cho x,y,z $\geq$ 0 thỏa x+y+z=3 và xy+yz+xz $\neq$ 0. CMR:
$\frac{x+1}{y+1}$ +$\frac{y+1}{z+1}$ +$\frac{z+1}{x+1}$ $\leq$ $\frac{25}{3\sqrt[3]{4.(xy+yz+xz)}}$
4. Cho a,b,c >0 thỏa $a^{3}$+$b^{3}$+$c^{3}$ =3. Tìm GTNN:
M= $\frac{a^2}{b}$ + $\frac{b^2}{c}$ + $\frac{c^2}{a}$
Giải thích các bước giải:
2.Ta có : $(a+b+c)^2\ge 3(ab+bc+ca)=3\rightarrow a+b+c\ge\sqrt{3}$
Ta có :
$b^2+1=\dfrac{3}{4}.(b^2+1)(\dfrac{1}{3}+1)\ge \dfrac{3}{4}(\dfrac{b}{\sqrt{3}}+1)^2$
$\rightarrow a\sqrt{b^2+1}\ge a.\sqrt{\dfrac{3}{4}(\dfrac{b}{\sqrt{3}}+1)^2}=\dfrac{\sqrt{3}}{2}.a(\dfrac{b}{\sqrt{3}}+1)$
Tương tự ta chứng minh được
$ b\sqrt{c^2+1}\ge \dfrac{\sqrt{3}}{2}.b(\dfrac{b}{\sqrt{3}}+1)$
$ c\sqrt{a^2+1}\ge \dfrac{\sqrt{3}}{2}.c(\dfrac{a}{\sqrt{3}}+1)$
$\rightarrow a\sqrt{b^2+1}+b\sqrt{c^2+1}+ c\sqrt{a^2+1}\ge \dfrac{\sqrt{3}}{2}.a(\dfrac{b}{\sqrt{3}}+1)+ \dfrac{\sqrt{3}}{2}.b(\dfrac{b}{\sqrt{3}}+1)+\dfrac{\sqrt{3}}{2}.c(\dfrac{a}{\sqrt{3}}+1)$
$\rightarrow a\sqrt{b^2+1}+b\sqrt{c^2+1}+ c\sqrt{a^2+1}\ge \dfrac{1}{2}(ab+bc+ca)+\dfrac{\sqrt{3}}{2}(a+b+c)$
$\rightarrow a\sqrt{b^2+1}+b\sqrt{c^2+1}+ c\sqrt{a^2+1}\ge \dfrac{1}{2}.1+\dfrac{\sqrt{3}}{2}.\sqrt{3}$
$\rightarrow a\sqrt{b^2+1}+b\sqrt{c^2+1}+ c\sqrt{a^2+1}\ge 2$