Question

Giúp mình với ạ😰😰😰

Answer 1

Đặt \(P=\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\)

\(P=1+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{ab}=1+\dfrac{a+b}{ab}+\dfrac{1}{ab}=1+\dfrac{2}{ab}\)

Với mọi a;b dương, ta có:

\(\left(a-b\right)^2\ge0\Leftrightarrow a^2+b^2\ge2ab\)

\(\Leftrightarrow a^2+2ab+b^2\ge4ab\Leftrightarrow\left(a+b\right)^2\ge4ab\)

\(\Rightarrow4ab\le1\Rightarrow ab\le\dfrac{1}{4}\Rightarrow\dfrac{1}{ab}\ge4\)

\(\Rightarrow P\ge1+2.4=9\) (đpcm)

Dấu "=" xảy ra khi \(a=b=\dfrac{1}{2}\)