a/Xét hiệu:
\(x+\dfrac{1}{x}\ge2\)
\(\Leftrightarrow x\ge2-\dfrac{1}{x}\)\(\Leftrightarrow x\ge\dfrac{2x-1}{x}\)
\(\Rightarrow x^2\ge2x-1\Leftrightarrow x^2-2x+1\ge0\Leftrightarrow\left(x-1\right)^2\ge0\)
(luôn đúng)
=> Đpcm
dấu ''='' xảy ra khi x = 1
b/ \(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\)
\(\Leftrightarrow\dfrac{a+b}{ab}\ge\dfrac{4}{a+b}\Leftrightarrow\left(a+b\right)^2\ge4ab\)
\(\Leftrightarrow a^2+2ab+b^2-4ab\ge0\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\Leftrightarrow\left(a-b\right)^2\ge0\)(luôn đúng)
dấu ''='' xảy ra khi a = b
Cách khác
a) x>0: \(x+\dfrac{1}{x}\ge2\sqrt{x.\dfrac{1}{x}}=2\)
\("="\Leftrightarrow x=\dfrac{1}{x};x>0\Rightarrow x=1\)
b) a;b>0 \(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{\left(1+1\right)^2}{a+b}=\dfrac{4}{a+b}\)
\("="\Leftrightarrow a=b\)