Докажите, что: (x+1/y)(y+1/x)>=4, если x>0, y>0.

\[\left( x + \frac{1}{y} \right)\left( y + \frac{1}{x} \right) \geq 4;\ \ \ \ x > 0;\ \ \]

\[y > 0\]

\[\left( \frac{xy + 1}{y} \right) \cdot \left( \frac{xy + 1}{x} \right) - 4 \geq 0\]

\[\frac{(xy + 1)^{2} - 4xy}{\text{xy}} \geq 0\]

\[\frac{x^{2}y^{2} + 2xy + 1 - 4xy}{\text{xy}} \geq 0\]

\[\frac{(xy - 1)^{2}}{\text{xy}} \geq 0 \Longrightarrow верно.\]

\[(xy - 1)^{2} \geq 0\ \ всегда;\ \ \ \]

\[xy > 0 - из\ условия.\]