a;b;c dương thỏa mãn $a+b+c=6$.Tìm min
$\frac{\sqrt{a^{2}+ab+b^{2}}}{bc+4}+\frac{\sqrt{b^{2}+bc+c^{2}}}{ca+4}+\frac{\sqrt{c^{2}+ca+a^{2}}}{ab+4}$
Cách giải sơ cấp cho bạn:(dùng cô-si hết nhé)
$\sum\frac{\sqrt{a^2+ab+b^2}}{bc+4}\geq\sum\frac{\sqrt3(a+b)}{2(bc+4)}\geq\sum\frac{\sqrt3(a+b)}{2(\frac{(b+c)^2}{4}+4)}$
Đến đây: đổi biến $(a+b,b+c,c+a)\rightarrow (4x,4y,4z)$
$4\sum x=2\sum a=12\Rightarrow \sum x=3$
Khi đó, ta cần chứng minh:
$\sum\frac{4\sqrt3x}{2(4y^2+4)}=\frac{\sqrt3}{2}\sum\frac{x}{y^2+1}=\frac{\sqrt3}{2}A$
$3-A=\sum x-\sum\frac{x}{y^2+1}=\sum\frac{xy^2}{y^2+1}\leq\sum\frac{xy}{2}\leq\frac{(\sum x)^2}{6}=\frac{3}{2}$
$\Rightarrow A\geq\frac{3}{2}$
Từ đó suy ra đpcm.
Dễ dàng chứng minh $a^2+ab+b^2 \ge \frac 34(a+b)^2$
Cmtt
$\Rightarrow VT \ge \sqrt{\frac 34}\left(\frac{a+b}{bc+4} +\frac{b+c}{ca+4}+\frac{c+a}{ab+4}\right)=\sqrt{\frac 34}.A$
Dùng bất đẳng thức hoán vị, ta có:
$A=\left(\frac{a}{bc+4} +\frac{b}{ca+4}+\frac{c}{ab+4}\right)+\left(\frac{b}{bc+4} +\frac{c}{ca+4}+\frac{a}{ab+4}\right)$
$\ge \left(\frac{c}{bc+4}+\frac{a}{ca+4}+\frac{b}{ab+4}\right)+\left(\frac{b}{bc+4} +\frac{c}{ca+4}+\frac{a}{ab+4}\right)$
$=\frac{b+c}{bc+4}+\frac{c+a}{ca+4}+\frac{a+b}{ab+4} =S$
Đến đây dùng bdt chebychev :$S \ge \frac 13[(b+c)+(c+a)+(a+b)\left[ \frac{1}{bc+4} +\frac{1}{ca+4}+\frac{1}{ab+4}\right]$
$ \ge 4.\frac{9}{ab+bc+ca+12} \ge \frac 32$3
Suy ra $VT \ge \sqrt{\frac 34}.\frac 32$
Suy ra $\min=\sqrt{\frac 34}.\frac 32$ khi $a=b=c=2$

bạn tự search gg nha,ở đây không tiện nói –  Aerilate 16-07-16 10:19 AM
cm như thế nào –  Lionel Messi 16-07-16 10:18 AM
Bđt hoán vị là j vậy –  Lionel Messi 16-07-16 10:17 AM

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