Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:
\(VT=\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}=\frac{a+b+c}{2}=VP\)
Đẳng thức xảy ra khi a = b = c
Bất đẳng thức Bunhiacopxki
B1: Cho a,b,c thỏa mãn: a+b+c=1. CMR: \(a^2+b^2+c^2\ge\dfrac{1}{3}\)
B2: Cho a,b,c dương thỏa mãn: \(a^2+4b^2+9c^2=2015\). CMR: \(a+b+c\le\dfrac{\sqrt{14}}{6}\)
B3: Cho a,b dương thỏa mãn: \(a^2+b^2=1\).CMR: \(a\sqrt{1+a}+b\sqrt{1+b}\le\sqrt{2+\sqrt{2}}\)
Cho a,b,c>0. CMR
\(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{c+a}\ge\dfrac{a+b+c}{2}\)
1. CMR: \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{a}{c}+\frac{c}{b}+\frac{b}{a}\)
2. Cho a, b , c >0 .CMR: \(\frac{bc}{a}+\frac{ac}{b}+\frac{ba}{c}\ge a+b+c\)
cho a,b,c>0,a+b+c=3
CMR P=\(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge3\)
Cho a,b,c>0. CMR
\(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge a+b+c\)
CMR : a,b,c >0
1,\(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{a+b}\dfrac{>}{ }\dfrac{a+b+c}{2}\)
2,\(\dfrac{a+b}{a^2+b^2}+\dfrac{b+c}{b^2+c^2}+\dfrac{a+c}{a^2+c^2}\dfrac{< }{ }\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
cho a,b,c > 0 .Cmr:
\(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\)
Cho a, b, c > 0
Cmr: \(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\)
Cho a,b,c là các số thực dương thỏa mãn a+b+c=1 . CMR \(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge3\left(a^2+b^2+c^2\right)\)
