Question

Cho \(a^3+b^3=2\) . Chmr: \(a+b\le2\)

Answer 1

\(a^3+b^3=2>0\Rightarrow a^3>-b^3\)

\(\Rightarrow a>-b\Leftrightarrow a+b>0\)

Giả sử \(a+b>2\Rightarrow\left(a+b\right)^3>8\)

\(\Leftrightarrow a^3+b^3+3ab\left(a+b\right)>8\)

\(\Leftrightarrow2+3ab\left(a+b\right)>8\)

\(\Rightarrow ab\left(a+b\right)>2\)

\(\Leftrightarrow ab\left(a+b\right)>a^3+b^3\)

\(\Leftrightarrow ab>a^2-ab+b^2\)

\(\Leftrightarrow\left(a-b\right)^2< 0\) \(\left(\text{vô lí}\right)\)

Vậy \(a+b\le2\)