\(\sqrt{a+b}.\sqrt{\frac{1}{a}+\frac{1}{b}}=\sqrt{\left(a+b\right)\left(\frac{1}{a}+\frac{1}{b}\right)}\)
\(=\sqrt{2+\frac{a}{b}+\frac{b}{a}}\ge\sqrt{2+2\sqrt{\frac{a}{b}.\frac{b}{a}}}=\sqrt{2+2}=2\)
Dấu bằng xảy ra khi a = b.
a,b,c>0 \(\frac{1}{\sqrt[3]{a+2b}}\) +\(\frac{1}{\sqrt[3]{b+2c}}\) +\(\frac{1}{\sqrt[3]{c+2a}}\) tim gtnn
cho a,b,c > 0 thỏa mãn \(a+b+c\le\frac{3}{2}\)
Tìm GTNN của \(A=\sqrt{a^2+\frac{1}{b^2}}+\sqrt{b^2+\frac{1}{c^2}}+\sqrt{c^2+\frac{1}{a^2}}\)
cho a,b>0 (a+b<=1). tim GTNN cua M=a+b+\(\frac{1}{a}+\frac{1}{b}\)
cho a,b>0(a+b<=1) tim GTNN cua J=\(\frac{1}{a^2+b^2}+\frac{1}{ab}\)
1/cho số a >0 tìm GTNN của P = 2a +\(\frac{4}{a}\)+\(\frac{16}{a+2}\)
2/ cho a,b,c là số thực ϵ [0;\(\frac{1}{4}\)) chứng minh:
\(\sqrt{a\left(1-4a\right)}+\sqrt{b\left(1-4b\right)}+\sqrt{c\left(1-4c\right)}\le\frac{3}{4}\)
3/ cho các số dương a,b,c tỏa abc = 1. Chứng minh
\(\frac{1}{a^2c+b^2c+1}+\frac{1}{b^2a+c^2a+1}+\frac{1}{c^2b+a^2b+1}\le1\)
a,b,c>0 , a+b+c=4
tim GTNN cua P=\(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\)
cho a,b,c > 0 thỏa mãn abc =1. Cmr: \(\frac{1}{\sqrt{ab+a+2}}+\frac{1}{\sqrt{bc+b+2}}+\frac{1}{\sqrt{ca+c+2}}\le\frac{3}{2}\)
Cho a, b, c > 0 thỏa mãn abc = 1. Tìm GTLN
P = \(\frac{1}{\sqrt{a^5+b^2+ab+6}}+\frac{1}{\sqrt{b^5+c^2+bc+6}}\frac{1}{\sqrt{c^5+a^2+ac+6}}\)
Cho a,b,c>0 thoả mãn a2+b2+c2=1
CMR: \(\frac{a^2+ab+1}{\sqrt{a^2+3ab+c^2}}+\frac{b^2+bc+1}{\sqrt{b^2+3bc+a^2}}+\frac{c^2+ca+1}{\sqrt{c^2+3ac+b^2}}\ge\sqrt{5}\left(a+b+c\right)\)