Ta có: \(\frac{ab}{a+b}\le\frac{a+b}{4};\frac{bc}{b+c}\le\frac{b+c}{4};\frac{ca}{c+a}\le\frac{c+a}{4}\)
\(\Rightarrow VT\le\frac{a+b+c}{2}\). Dấu = xảy ra khi a = b= c mà b =2 => a = c =2
\(A=\frac{a^2+bc}{b+ac}+\frac{b^2+ca}{c+ab}+\frac{c^2+ab}{a+bc}\)
\(=\frac{3\left(a^2+bc\right)}{\left(a+b+c\right)b+3ac}+\frac{3\left(b^2+ca\right)}{\left(a+b+c\right)c+3ab}+\frac{3\left(c^2+ab\right)}{\left(a+b+c\right)a+3bc}\)
\(\ge\frac{3\left(a^2+bc\right)}{\left(a^2+bc\right)+\left(b^2+ca\right)+\left(c^2+ab\right)}+\frac{3\left(b^2+ca\right)}{\left(a^2+bc\right)+\left(b^2+ca\right)+\left(c^2+ab\right)}+\frac{3\left(c^2+ab\right)}{\left(a^2+bc\right)+\left(b^2+ca\right)+\left(c^2+ab\right)}=3\)
\(A=\frac{a^2}{a+bc}+\frac{b^2}{b+ca}+\frac{c^2}{c+ab}\)cho ab+bc+ca=abc và a,b,c>0 Tìm min
Bài 1:Cho a,b,c>0. Chứng minh rằng:
\(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\)
Bài 2: Cho 3 số dương a,b,c. Tìm giá trị nhỏ nhất của biểu thức:
\(A=\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right)+a^2+b^2+c^2-ab-bc-ca+2020\)
1,Cho a,b,c>0 thỏa mãn a+b+c=abc.CMR:
\(\frac{bc}{a\left(1+bc\right)}+\frac{ca}{b\left(1+ca\right)}+\frac{ab}{c\left(1+ab\right)}\ge\frac{3\sqrt{3}}{4}\)
2,Cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=3\)
Tìm GTLN của P= \(\sqrt{\frac{a^2}{a^2+b+c}}+\sqrt{\frac{b^2}{b^2+c+a}}+\sqrt{\frac{c^2}{c^2+a+b}}\)
3,Cho a,b,c>0 thỏa mãn a+b+c=3.
Tìm GTLN của Q= \(2\sqrt{abc}\left(\frac{1}{\sqrt{3a^2+4b^2+5}}+\frac{1}{\sqrt{3b^2+4c^2+5}}+\frac{1}{\sqrt{3c^2+4a^2+5}}\right)\)
4,Cho a,b,c>0.
Tìm GTLN của P= \(\frac{\sqrt{ab}}{c+3\sqrt{ab}}+\frac{\sqrt{bc}}{a+3\sqrt{bc}}+\frac{\sqrt{ca}}{b+3\sqrt{ca}}\)
cho a,b,c là số thực dương. Cmr:
\(\frac{a}{b^2+bc+c^2}+\frac{b}{c^{^2}+ca+a^2}+\frac{c}{a^2+ab+b^2}\ge\frac{a+b+c}{ab+bc+ca}\)
Cho ab+bc+ca=abc ; a,b,c >0 Tìm min \(A=\frac{a^2}{a+bc}+\frac{b^2}{b+ac}+\frac{c^2}{c+ab}\)
a,b,c>0 tìm min
\(P=\frac{a^3+b^3+c^3}{2abc}+\frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}\)
nhanh giúp với!
đặt \(P=\frac{1}{1-ab}+\frac{1}{1-bc}+\frac{1}{1-ca}\)
\(\Rightarrow P-3=\frac{ab}{1-ab}+\frac{bc}{1-bc}+\frac{ca}{1-ca}\le\frac{ab}{1-\frac{a^2+b^2}{2}}+\frac{bc}{1-\frac{b^2+c^2}{2}}+\frac{ca}{1-\frac{c^2+a^2}{2}}\)
\(\le\frac{1}{2}.\frac{\left(a+b\right)^2}{\left(a^2+c^2\right)+\left(b^2+c^2\right)}+\frac{1}{2}.\frac{\left(b+c\right)^2}{\left(a^2+b^2\right)+\left(c^2+a^2\right)}+\frac{1}{2}.\frac{\left(c+a\right)^2}{\left(b^2+c^2\right)+\left(b^2+a^2\right)}\)
\(\le\frac{1}{2}.\left(\frac{a^2}{a^2+c^2}+\frac{b^2}{b^2+c^2}+\frac{b^2}{a^2+b^2}+\frac{c^2}{c^2+a^2}+\frac{c^2}{b^2+c^2}+\frac{a^2}{b^2+a^2}\right)=\frac{3}{2}\)
\(\Rightarrow P-3\le\frac{3}{2}\Rightarrow P\le\frac{9}{2}\)
biết \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=3\)tìm gtnn của \(P=\frac{ab^2}{a+b}+\frac{bc^2}{b+c}+\frac{ca^2}{c+a}\)
