# Prove that $x+\frac{1}{x}\geq2$ [duplicate]

To prove $x+\frac{1}{x}\geq2$ where $x$ is a positive real number. This is what i try: $$\text{We need to prove } \hspace{1cm} x+\frac{1}{x}-2\geq 0$$ now,$$\frac{x^2-2x+1}{x}=(x-2)+\frac{1}{x}$$ its enough to show that $$\frac{1}{x}\geq(x-2)\hspace{0.5cm} \text{ when } \hspace{0.2cm}0<x\leq 2$$ we can easily show it using the graph.

but my question is can we do it algebraically or using calculus to prove it without any reference to the graphs.

## marked as duplicate by Martin R, E. Joseph, Rohan, Community♦Jan 26 '17 at 9:14

• It looks like you went from $A$ to $B$ and then came back to $A$! – NeedForHelp Jan 26 '17 at 9:11
• Just solve this inequality. $$\left(\sqrt{x}+\frac{1}{\sqrt x}\right)^2\ge0$$ – Harsh Kumar Jan 26 '17 at 12:36
$\dfrac{x+\dfrac{1}{x}}{2} \ge \sqrt{x \times \dfrac{1}{x} }$
$\dfrac{x+\dfrac{1}{x}}{2} \ge 1$
$x+\dfrac{1}{x} \ge 2$