Proving x+1/x >= 2 for x>0 in Math: Helpful Tips and Tricks

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In summary, the conversation discusses how to prove the inequality x+1/x >= 2 for all x>0. The participants mention multiplying through the inequality and using the fact that A.M. >= G.M. to arrive at the desired result.
  • #1
mercury
ok i have a really stupid problem ...but i can't seem to get it...would appreciate any help...
how do u prove that x+1/x >= 2 for all x>0...
is the above state ment true in the first place..i think it is...but can't seem to prove it...i keep coming to the fact that it holds for x>1 and equality holds for x=1 but can't seem to prove it...obviously for 0<x<1...
 
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  • #2
Since you are specifying that x> 0, you can multiply through the inequality by x without any problem:

You want to get x+ 1/x>= 2 and, multiplying by x, that is the same as x2+ 1>= 2x.

That, of course, is the same as x2- 2x+ 1>= 0 and you KNOW that's true.

Well, HOW do you know it's true? Also remember you have to ARRIVE at "x+ 1/x>= 2", not start there. Start with what you KNOW is true and work backwards.
 
  • #3
Another method is to use the fact that A.M. > = G.M. as x > 0

Where AM stands for arithmatic mean and GM stands for Geometric mean.
 

1. How can I prove the inequality x+1/x >= 2 for x>0 in math?

To prove this inequality, you can use the AM-GM inequality, which states that for any set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean. In this case, we can set x and 1/x as our two positive numbers and use the AM-GM inequality to prove the given inequality.

2. What is the AM-GM inequality in math?

The AM-GM inequality, or the Arithmetic Mean - Geometric Mean inequality, states that for any set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean. In other words, the average of a set of numbers is always greater than or equal to the product of those numbers.

3. Can I use induction to prove x+1/x >= 2 for x>0?

No, induction is not a suitable method to prove this inequality. Induction is typically used to prove statements about integers or discrete values, but in this case, x is a real number. Moreover, induction requires a base case, which is not applicable in this scenario.

4. Are there other methods to prove x+1/x >= 2 for x>0?

Yes, there are other methods to prove this inequality, such as using calculus or algebraic manipulation. However, the AM-GM inequality is the simplest and most efficient method for this particular case.

5. Can I prove x+1/x >= 2 for x>0 without using the AM-GM inequality?

Yes, it is possible to prove this inequality using other methods. However, as mentioned before, the AM-GM inequality is the most straightforward and efficient method for this case. Using other methods may require more complex and lengthy proofs.

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