Question

Cho a, b dương. CMR:

\(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)

Answer 1

https://i.imgur.com/7Redyx8.jpg

Answer 2

\(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}-\frac{4}{a+b}\ge0\)

\(\Leftrightarrow\frac{b\left(a+b\right)+a\left(a+b\right)-4ab}{ab\left(a+b\right)}\ge0\)

\(\Leftrightarrow b^2+ab+a^2+ab-4ab\ge0\)

\(\Leftrightarrow\left(a-b\right)^2\ge0\forall a,b\)

\(\Rightarrow\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)

Answer 3

Sao không ai dùng AM-GM nhỉ?

Áp dụng BĐT AM-GM ta có:

\(\frac{1}{a}+\frac{1}{b}\ge\frac{2}{\sqrt{ab}}=\frac{2}{\frac{a+b}{2}}=\frac{4}{a+b}\)

Dấu "= " xảy ra <=> a=b