Chắc là đề yêu cầu tìm GTNN của P
\(P=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{a^2}{ab+ac}+\frac{b^2}{ab+bc}+\frac{c^2}{ac+bc}\)
\(P\ge\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ca\right)}\ge\frac{3\left(ab+bc+ca\right)}{2\left(ab+bc+ca\right)}=\frac{3}{2}\)
Dấu "=" xảy ra khi \(a=b=c\)
1, cho a,b,c ≥0 chứng minh các bất đẳng thức sau:
a, (a+b)(b+c)(c+a) ≥ 8abc
b, \(\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\ge a+b+c,vớia+b+c>0\)
c, \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}vớia,b,c>0\)
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: \(\frac{bc}{a}+\frac{ac}{b}+\frac{ba}{c}\ge a+b+c\)
1) \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}\)
2) với \(\left\{{}\begin{matrix}a,b,c>0\\a+b+c=3\end{matrix}\right.\) chứng minh \(\frac{a^3}{b\left(2c+a\right)}+\frac{b^3}{c\left(2a+b\right)}+\frac{c^3}{a\left(2b+c\right)}\ge1\)
cho \(c\ge b\ge a>0\) . Cmr: \(\frac{2a^2}{b+c}+\frac{2b^2}{c+a}+\frac{2c^2}{a+b}\le\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
CM với \(\forall a,b,c>0\) thì \(\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{b+a}\ge\frac{a+b+c}{2}\)
cho a,b,c > 0 . Cmr:
\(\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\ge\frac{a+b+c}{3}\)
cho a,b,c >0 và a+b+c=3 . cmr :
\(\frac{a}{\sqrt{b+c+2}}+\frac{b}{\sqrt{a+c+2}}+\frac{c}{\sqrt{a+b+2}}\ge\frac{3}{5}\)
cho a,b,c > 0 thỏa mãn a+b+c = 3. Cmr:
\(\frac{a^3}{b^2+c^2}+\frac{b^3}{c^2+a^2}+\frac{c^3}{a^2+b^2}\ge\frac{3}{2}\)
cho a,b,c>0 cm
\(\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}\ge a+b+c\)
