Ta có : \(\left(a-b\right)^2\ge0\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\)
\(\Leftrightarrow a^2+b^2\ge2ab\)
\(\Leftrightarrow a^2+b^2+2ab\ge4ab\)
\(\Leftrightarrow\left(a+b\right)^2\ge4ab\)
\(\Leftrightarrow\dfrac{a+b}{ab}=\dfrac{4}{a+b}\)
\(\Leftrightarrow=\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\left(dpcm\right)\)
Ta có: (a-b)2 \(\ge\) 0 (dấu "=" xảy ra khi a=b)
\(\Leftrightarrow\left(a+b\right)^2-4ab\ge0\)
\(\Leftrightarrow\left(a+b\right)^2\ge4ab\)
\(\Leftrightarrow\dfrac{a+b}{ab}\ge\dfrac{4}{a+b}\) (vì a,b > 0 )
\(\Leftrightarrow\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\) . Dấu "=" xảy ra khi a=b.
