Question

Answer 1

\(a,b>0\)

\(2ab-4=a+b>0\Rightarrow2ab>4\Rightarrow ab>2\).

\(2ab-4=a+b\)

\(\Leftrightarrow\left(2ab-4\right)^2=\left(a+b\right)^2\)

-Mà \(\left(a+b\right)^2\ge4ab\) (áp dụng BĐT AM-GM).

\(\Rightarrow\left(2ab-4\right)^2\ge4ab\)

\(\Leftrightarrow4a^2b^2-16ab+16\ge4ab\)

\(\Leftrightarrow4a^2b^2-20ab+16\ge0\)

\(\Leftrightarrow\left(2ab-5\right)^2-9\ge0\)

\(\Leftrightarrow\left(2ab-8\right)\left(2ab-2\right)\ge0\).

\(\Leftrightarrow2ab-8\ge0\) (do \(2ab-2>0\)).

\(\Leftrightarrow ab\ge4\)

\(\left(ab\right)=4\Leftrightarrow a=b=2\)
-Vậy \(\left(ab\right)_{min}=4\)