Question

1. cho a,b,c là các số dương  .Cmr :

\(\dfrac{a^3+b^3}{ab}+\dfrac{b^3+c^3}{bc}+\dfrac{a^3+c^3}{ac}\ge2\left(a+b+c\right)\)

Answer 1

Chứng minh: \(x^3+y^3\ge xy\left(x+y\right)\left(1\right)\)

\(x^3+y^3\ge xy\left(x+y\right)\)

\(\Leftrightarrow\left(x+y\right)^3-3xy\left(x+y\right)\ge xy\left(x+y\right)\)

\(\Leftrightarrow\left(x+y\right)^3\ge4xy\left(x+y\right)\)

\(\Leftrightarrow\left(x+y\right)^2\ge4xy\)

\(\Leftrightarrow\left(x-y\right)^2\ge0\) đúng

\(\Rightarrow\left(1\right)\) đúng

Áp dụng BĐT \(x^3+y^3\ge xy\left(x+y\right)\)

\(\dfrac{a^3+b^3}{ab}+\dfrac{b^3+c^3}{bc}+\dfrac{c^3+a^3}{ca}\)

\(\ge\dfrac{ab\left(a+b\right)}{ab}+\dfrac{bc\left(b+c\right)}{bc}+\dfrac{ca\left(c+a\right)}{ca}\)

\(=2\left(a+b+c\right)\)