Question

[Nguyễn Lê Quân]

chứng minh rằng 

1/a + 1/b +1/c >= 9 với mọi a , b , c>0 và a+b+c=1

Answer 1

[Đinh Thùy Linh]

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{a+b+c}{a}+\frac{a+b+c}{b}+\frac{a+b+c}{c}=1+\frac{b}{a}+\frac{c}{a}+1+\frac{a}{b}+\frac{c}{b}+1+\frac{a}{c}+\frac{b}{c}.\)

\(=3+\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)+\left(\frac{a}{c}+\frac{c}{a}\right)\)

Theo Cosy với a;b;c >0

 \(\frac{a}{b}+\frac{b}{a}\ge2\sqrt{\frac{a}{b}\cdot\frac{b}{a}}=2\);\(\frac{b}{c}+\frac{c}{b}\ge2\sqrt{\frac{b}{c}\cdot\frac{c}{b}}=2\);\(\frac{a}{c}+\frac{c}{a}\ge2\sqrt{\frac{a}{c}\cdot\frac{c}{a}}=2\)

Do đó: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge3+2+2+2=9\)đpcm.

Dấu "=" khi a=b=c=1/3.