Chứng minh bất đẳng thức:

\(\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right) \ge \frac{9}{{x + y + z}}\)  (dấu bằng xảy ra khi x = y = z).

Xét \(\left( {x + y + z} \right)\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)\)

\( = \frac{x}{x} + \frac{x}{y} + \frac{x}{z} + \frac{y}{x} + \frac{y}{y} + \frac{y}{z} + \frac{z}{x} + \frac{z}{y} + \frac{z}{z}\)

\( = 3 + \left( {\frac{x}{y} + \frac{y}{x}} \right) + \left( {\frac{x}{z} + \frac{z}{x}} \right) + \left( {\frac{y}{z} + \frac{z}{y}} \right)\)

\( \ge 3 + 2\sqrt {\frac{x}{y}\,.\,\frac{y}{x}} + 2\sqrt {\frac{x}{z}\,.\,\frac{z}{x}} + 2\sqrt {\frac{x}{y}\,.\,\frac{y}{x}} \) (với x, y, z > 0)

\( = 3 + 2 + 2 + 2 = 9\)

Vậy \(\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right) \ge \frac{9}{{x + y + z}}\)

Dấy “=” xảy ra khi:

\(\left\{ \begin{array}{l}\frac{x}{y} = \frac{y}{x}\\\frac{z}{x} = \frac{x}{z}\\\frac{y}{z} = \frac{z}{y}\end{array} \right. \Rightarrow x = y = z\)