Question

Cho \(a\ge3;a+b\ge5\). CMR : \(a^2+b^2\ge13\)

Answer 1

theo bđt bunhi a cốp xiki ta có\(\left(3a+2b\right)^2\le\left(3^2+2^2\right)\left(a^2+b^2\right)\le13\left(a^2+b^2\right)\)

mặt khác; theo giả thiết:

\(\left(3a+2b\right)^2=\left(10+a\right)^2\ge13^2\)

từ đó suy ra a^2+b^2>=13

Answer 2

Ta có : \(\left\{{}\begin{matrix}a\ge3\\a+b\ge5\end{matrix}\right.\Rightarrow b\ge2\)

\(\Rightarrow ab\ge6\Rightarrow2ab\ge12\)

\(a^2+b^2=\left(a+b\right)^2-2ab\ge5^2-12=13\) ( đpcm )