Câu hỏi của ๖ۣۜDũ๖ۣۜN๖ۣۜG

Question

Cho 3 số dương a,b,c thỏa mãn a+b+c = 6.CMR

$\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{b+c}}+\frac{1}{\sqrt{c+a}}\ge\frac{3}{2}$

Nguyễn Việt Lâm · Accepted Answer

$VT=\frac{4}{2.2\sqrt{a+b}}+\frac{4}{2.2\sqrt{b+c}}+\frac{4}{2.2\sqrt{c+a}}$

$VT\ge\frac{4}{a+b+4}+\frac{4}{b+c+4}+\frac{4}{c+a+4}$

$VT\ge\frac{36}{a+b+4+b+c+4+c+a+4}=\frac{36}{24}=\frac{3}{2}$

Dấu "=" xảy ra khi $a=b=c=2$Câu trả lời: $VT=\frac{4}{2.2\sqrt{a+b}}+\frac{4}{2.2\sqrt{b+c}}+\frac{4}{2.2\sqrt{c+a}}$

$VT\ge\frac{4}{a+b+4}+\frac{4}{b+c+4}+\frac{4}{c+a+4}$

$VT\ge\frac{36}{a+b+4+b+c+4+c+a+4}=\frac{36}{24}=\frac{3}{2}$

Dấu "=" xảy ra khi $a=b=c=2$

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