Question

[Phương Minh Trần]

Cho a,b,c>0

Cm(a+b+c)(\( {1\over a}\)+\({1 \over b}\)+\({1\over c}\))>=9

Giúp minh vơiz 🙂😊🤗

Answer 1

[Đinh quang hiệp]

vì a;b;c >0 nên 1/a;1/b;1/c>0

\(\Rightarrow\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)>=3\sqrt[3]{abc}\cdot3\sqrt[3]{\frac{1}{a}\cdot\frac{1}{b}\cdot\frac{1}{c}}\)(bđt cosi)

\(=3\sqrt[3]{abc}\cdot3\cdot\frac{1}{\sqrt[3]{abc}}=9\cdot\sqrt[3]{abc}\cdot\frac{1}{\sqrt[3]{abc}}=9\cdot\frac{\sqrt[3]{abc}}{\sqrt[3]{abc}}=9\)

\(\Rightarrow\)đpcm

Answer 2

[Không Tên]

cách khác nhé:

\(VT=\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

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

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

C/m BĐT phụ:    \(\frac{x}{y}+\frac{y}{x}\ge2\)      (x,y > 0)

               \(\Leftrightarrow\)\(\frac{x^2}{xy}+\frac{y^2}{xy}\ge\frac{2xy}{xy}\)

              \(\Leftrightarrow\) \(\frac{x^2+y^2-2xy}{xy}\ge0\)

             \(\Leftrightarrow\) \(\frac{\left(x-y\right)^2}{xy}\ge0\)  luôn đúng

Dấu "=" xảy ra  \(\Leftrightarrow\) \(x=y\)

Áp dụng BĐT trên ta có:

      \(VT=3+\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{a}{c}+\frac{c}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)\ge3+2+2+2=9\)

hay   \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\)  (đpcm)

Dấu  "="  xảy ra  \(\Leftrightarrow\)\(a=b=c\)