Question

Доведіть, що (x+1/y)(y+1/x)≥4, якщо x>0, y>0

Answer 1

Ответ:

$\displaystyle (x+\frac{1}{y})(y+\frac{1}{x})\ge 4$

Объяснение:

формула:

$\displaystyle (a-b)^2\ge0\\\\a^2-2ab+b^2\ge0\ \ \ |+4ab\\\\a^2+2ab+b^2\ge 4ab\\\\(a+b)^2\ge 4ab\ \ \ |\sqrt{}\\\\a+b\ge 2\sqrt{ab}$

$\displaystyle (x+\frac{1}{y})(y+\frac{1}{x})\ge 2\sqrt{x\cdot \frac{1}{y}}\cdot 2\sqrt{y\cdot \frac{1}{x}}\\\\(x+\frac{1}{y})(y+\frac{1}{x})\ge 4\sqrt{ \frac{x}{y}}\cdot \sqrt{\frac{y}{x}}\\\\(x+\frac{1}{y})(y+\frac{1}{x})\ge 4\sqrt{ \frac{x}{y}\cdot \frac{y}{x}}\\\\(x+\frac{1}{y})(y+\frac{1}{x})\ge 4\sqrt{1}\\\\(x+\frac{1}{y})(y+\frac{1}{x})\ge 4\cdot 1\\\\(x+\frac{1}{y})(y+\frac{1}{x})\ge 4$