a)\(\left(a+\frac{b}{2}\right)^2\ge ab\)
\(\Leftrightarrow a^2+ab+\frac{b^2}{4}\ge ab\)
\(\Leftrightarrow a^2+ab+\frac{b^2}{4}-ab\ge0\)
\(\Leftrightarrow a^2+\frac{b^2}{4}\ge0\)(luôn lúng)
vậy \(\left(a+\frac{b}{2}^2\right)\ge ab\)
b)\(\frac{a}{b}+\frac{b}{a}\ge2\)
\(\Leftrightarrow\frac{a^2+b^2}{ab}-2\ge0\)
\(\Leftrightarrow\frac{a^2+b^2+2ab}{ab}\ge0\)
\(\Leftrightarrow\frac{\left(a+b\right)^2}{ab}\ge0\)(luôn đóng vì a,b>0)
Vậy \(\frac{a}{b}+\frac{b}{a}\ge2\)với a,b>0
b) \(\frac{a}{b}\rightarrow x\).C/m: \(x+\frac{1}{x}\ge2\)
Có \(\left(\sqrt{x}-\sqrt{\frac{1}{x}}\right)^2\ge0\Rightarrow x-2+\frac{1}{x}\ge0\Rightarrow x+\frac{1}{x}\ge2\) (đpcm)
shitbo cái đó tự suy ra m,soi mói nhìn kĩ zô nhá!
a) \(\left(\frac{a+b}{2}\right)^2\ge ab\)
\(\Leftrightarrow\left(\frac{a+b}{2}\right)^2-ab\ge0\)
\(\Leftrightarrow a^2+2ab+b^2\ge4ab\Leftrightarrow\left(a-b\right)^2\ge0\)( đúng ) (đpcm)
đăng thức xảy ra \(\Leftrightarrow a=b\)
bn Đạt đúng
The Tempest
