How am I to prove this inequality without use of calculus: for any real x>0, x+1/x >= 2 ? Thanks for any help.
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possible duplicate of How to prove this inequality $ x + \frac{1}{x} \geq 2 $ – user26486 May 21 '15 at 00:16
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Hint: Use the Arithmetic Mean - Geometric Mean inequality.
$$\frac {x + y} 2 \ge \sqrt {xy}$$
for $x, y \ge 0$
GFauxPas
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AM-GM is $\frac{a_1+a_2+\cdots+a_k}{k}\ge \sqrt[k]{a_1a_2\cdots a_k}$, which is not so simple to prove and overkill here where we can simply complete the square, as shown in Alexey's answer. – user26486 May 21 '15 at 00:13
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The simple case of $\frac {x + y} 2 \ge \sqrt{xy}$ can be granted as given. I find this simpler. "Easier" is subjective. – GFauxPas May 21 '15 at 00:15
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And this way is easy to generalise to $n$ positive numbers and their inverses. – Bernard May 21 '15 at 00:17