Áp dụng bđt \(\frac{x^2}{m}+\frac{y^2}{n}+\frac{z^2}{p}\ge\frac{\left(x+y+z\right)^2}{m+n+p}\) được :
\(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}=\frac{a+b+c}{2}\) (đpcm)
CMR với a,b,c>0 thì \(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c\)
CMR: \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}\) với a,b,c > 0
Cho a,b,c > 0
CMR : \(\frac{a^2}{b^2}+\frac{b^2}{a^2}\ge\frac{a}{b}+\frac{b}{a}\)
Cho a,b,c>0.CM:
\(\frac{a}{b^2}+\frac{b}{c^2}+\frac{c}{a^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
2. Cho a,b,c>0. Chứng minh: \(\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\ge\frac{3}{2}\)
Cho a,b,c>0. Chứng minh: \(\frac{a^2}{b+3c}+\frac{b^2}{c+3a}+\frac{c^2}{a+3b}\ge\frac{a+b+c}{4}\)
Bài 1: cho \(a,b,c\ge0\) và a+b+c=1. Chứng minh rằng :
a,\(\left(1-a\right)\cdot\left(1-b\right)\cdot\left(1-c\right)\ge8\cdot a\cdot b\cdot c\)
b,\(16\cdot a\cdot b\cdot c\ge a+b\)
c,\(\frac{a}{1+a}+\frac{2\cdot b}{2+b}+\frac{3\cdot c}{3+c}\le\frac{6}{7}\)
Bài 2: cho a,b,c>0 và a.b.c=0 chứng minh rằng:
\(\frac{b\cdot c}{a^2\cdot b+a^2\cdot c}+\frac{a\cdot c}{b^2\cdot c+b^2\cdot a}+\frac{a\cdot b}{c^2\cdot a+c^2\cdot b}\ge\frac{3}{2}\)
cho a,b,c>0 chứng minh rằng:
1)
\(\frac{a^2}{b+c}\) +\(\frac{b^2}{c+a}+\) \(\frac{c^2}{a+c}\ge\frac{a+b+c}{2}\)
CMR nếu
\(c^2+2\left(ab-ac-bc\right)=0,b\ne c,a+b\ne c\) thì \(\frac{a^2+\left(a-c\right)^2}{b^2+\left(b-c\right)^2}=\frac{a-c}{b-c}\)