Chứng minh rằng: (x + y + z) (1/x + 1/y + 1/z) > = 9 (với x, y, z > 0)
Chứng minh rằng: \(\left( {x + y + z} \right)\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right) \ge 9\) (với x, y, z > 0).
\(VT = \left( {x + y + z} \right)\left( {\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} \right)\)
\( = \frac{x}{x} + \frac{x}{y} + \frac{x}{z} + \frac{y}{x} + \frac{y}{y} + \frac{y}{z} + \frac{z}{x} + \frac{z}{y} + \frac{z}{z}\)
\( = 3 + \left( {\frac{x}{y} + \frac{y}{x}} \right) + \left( {\frac{x}{z} + \frac{z}{x}} \right) + \left( {\frac{y}{z} + \frac{z}{y}} \right)\)
\( \ge 3 + 2\sqrt {\frac{x}{y}\,.\,\frac{y}{x}} + 2\sqrt {\frac{x}{z}\,.\,\frac{z}{x}} + 2\sqrt {\frac{x}{y}\,.\,\frac{y}{x}} \) (với x, y, z > 0)
\( = 3 + 2 + 2 + 2 = 9\)