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March 10th, 2013, 07:51 PM   #1
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proof involving real number!

Let x be a positive real number. Prove that if , then .
(a) a direct proof
Assume is true. Then,

Since and , is positive real number. If x is less than or equal to 2, then . Therefore, . Q.E.D

Any comment?? And can anyone prove this by contrapositive?? Thanks.
eChung00 is offline  
March 10th, 2013, 09:30 PM   #2
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Re: proof involving real number!

Correct. Maybe you should have stated the reason for x(x-2)<=2.

Anyway. x^2-x = (x-1/2)^2 - 1/4
you can wtite
(x-1/2)^2 > 9/4

As we can see the (in)equality x-2/x ? 1 is logically the same than x ? 2.
csak is offline  
March 12th, 2013, 02:41 PM   #3
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Re: proof involving real number!

Contrapositive: If then so .
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