Cho ab+bc+ca=1. Tìm gia trị nhỏ nhất của:\(P=\frac{a^8}{\left(a^4+b^4\right)\left(a^2+b^2\right)}+\frac{b^8}{\left(b^4+c^4\right)\left(b^2+c^2\right)}+\frac{c^8}{\left(c^4+a^4\right)\left(c^2+a^2\right)}\)
Bài 1 :Cho a,b,c dương thỏa mãn a+b+c=2
CMR \(\frac{bc}{\sqrt{3a^2+4}}+\frac{ca}{\sqrt{3b^2+4}}+\frac{ab}{\sqrt{3c^2+4}}\ge\frac{\sqrt{3}}{3}\)
Bài 2:Cho a,b,c>0. CMR
\(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\frac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)
Cho các số thực a, b, c thỏa mãn a+b+c=3. Tìm GTLN của biểu thức:\(P=3\left(ab+bc+ca\right)+\frac{1}{2}\left(a-b\right)^2+\frac{1}{4}\left(b-c\right)^2+\frac{1}{8}\left(c-a\right)^2\)
\(P=\frac{a}{\sqrt{\left(b+1\right)\left(b^2-b+1\right)}}+\frac{b}{\sqrt{\left(c+1\right)\left(c^2-c+1\right)}}+\frac{c}{\sqrt{\left(a+1\right)\left(a^2-a+1\right)}}\)
\(\ge\frac{2a}{b^2+2}+\frac{2b}{c^2+2}+\frac{2c}{a^2+2}=\left(a+b+c\right)-\left(\frac{ab^2}{b^2+2}+\frac{bc^2}{c^2+2}+\frac{ca^2}{a^2+2}\right)\)
\(=6-\left(\frac{2ab^2}{b^2+4+b^2}+\frac{2bc^2}{c^2+4+c^2}+\frac{2ca^2}{a^2+4+a^2}\right)\ge6-\left(\frac{2ab}{b+4}+\frac{2bc}{c+4}+\frac{2ca}{a+4}\right)\)
\(=6-\left(2a+2b+2c-\frac{8a}{b+4}-\frac{8b}{c+4}-\frac{8c}{a+4}\right)\)
\(=\frac{8a}{b+4}+\frac{8b}{c+4}+\frac{8c}{a+4}-6=\frac{8a^2}{ab+4a}+\frac{8b^2}{bc+4b}+\frac{8c^2}{ca+4c}-6\)
\(\ge\frac{8\left(a+b+c\right)^2}{\left(ab+bc+ca\right)+4\left(a+b+c\right)}-6\ge\frac{288}{\frac{\left(a+b+c\right)^2}{3}+24}-6=2\)
cho a,b,c là ác số dương thỏa mãn a^2 + b^2 +c^2 <= 3c
tìm gtnn A=\(\frac{1}{\left(a+1\right)^2}+\frac{8}{\left(b+3\right)^2}+\frac{4}{\left(c+2\right)}\)
Cho a,b,c là các số thực dương thỏa mãn a+b+c=3abc. Chứng minh rằng :
\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2\left[\frac{a^4}{\left(ab+1\right)\left(ac+1\right)}+\frac{b^4}{\left(bc+1\right)\left(ab+1\right)}+\frac{c^4}{\left(ca+1\right)\left(bc+1\right)}\right]\ge\frac{27}{4}\)
Cho a,b,c > 0 thỏa mãn a + b + c = 3.
Chứng minh rằng: \(\frac{a^4}{\left(b+c\right)\left(b^2+c^2\right)}+\frac{b^4}{\left(c+a\right)\left(c^2+a^2\right)}+\frac{c^4}{\left(a+b\right)\left(a^2+b^2\right)}\ge\frac{3}{4}\)
cho a,b,c>0 thỏa mãn \(a^4+b^4+c^4\le3\)
CMR
\(\frac{a^2}{c\left(a+b\right)^3}+\frac{b^2}{a\left(b+c\right)^3}+\frac{c^2}{^{b\left(c+a\right)^3}}\ge\frac{3}{8}\)
Cho số thực dương a,b,c thỏa mãn abc =1 . Tìm GTNN của biểu thức
P = \(\frac{\left(1+a\right)^2+b^2+5}{ab+a+4}+\frac{\left(1+b\right)^2+c^2+5}{bc+b+4}+\frac{\left(1+c\right)^2+a^2+5}{ac+c+4}\)