How do I prove that x + 1/x is greater than or equal to 2 if x > 0

[JasonJo]

How to prove this?

How do I prove that x + 1/x is greater than or equal to 2 if x > 0

i'm not allowed to use calculus either.

i got that x + 1/x is greater than zero, but i can't get greater than or equal to 2.

 

[0rthodontist]

Assume for contradiction that x > 0 yet x + 1/x < 2. Transform it into a quadratic and show that this is impossible.

 

[George Jones]

Hint: (x - 1)^2 > 0.

Regards,
George

 

[pivoxa15]

First make the equation into x^2-2x+1 >= 0 by rearranging and multiplying by x.

You should find it is (x-1)^2 so for any x (not just x>0) this function is greater than or equal to 0 because of ^2.

For your situation it is obvious that x>0 which is certainly true for the factorised function. Therefore you have proved that x+1/x is greater than or equal to 2.

 

[JasonJo]

First make the equation into x^2-2x+1 >= 0 by rearranging and multiplying by x.

You should find it is (x-1)^2 so for any x (not just x>0) this function is greater than or equal to 0 because of ^2.

For your situation it is obvious that x>0 which is certainly true for the factorised function. Therefore you have proved that x+1/x is greater than or equal to 2.

before i posted this, i did get the algebraic manipulation of (x-1)^2, but i thought that only proved it for x greater than or equal to 1. but i guess since it's an equivalent statement, it's the same thing.

thanks everyone

 

[vaishakh]

Another method is arithmetis-geometric mean inequality.

 

[HallsofIvy]

Strictly speaking a proof would work the other way:

for any x, $(x-1)^2\ge 0$ so $x^2- 2x+ 1\ge 0$.
Adding 2x to both sides, $x^2+ 1\ge 2x$. Finally, dividing both sides by the positive number x, $x+ \frac{1}{x}\ge 2$