If x and y are positive, is x + y 2xy

Question

If x and y are positive, is  x+y>2\sqrt{xy} ?

(1)  x>y 
(2)  (x-y)^2>0

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Nozlatling · Answer

This question is about an interesting fact about the arithmetic mean (AM) and the geometric mean (GM).

If you are familiar with the concept of AM and GM, this is a straightforward one; Both 1 and 2 alone are sufficient because both say that x is not equal to y.

Here's the proof.
AM: (x+y)/2
GM: √xy
AM must always be equal to or greater than GM for following reason, and this is identical to the theme of this question.
(what you want to know is whether x and y suffice the certain condition where AM is NOT equal to but greater than GM.)

If AM>GM,
→(x+y)/2>√xy
→ x+y>2√xy (This is the question)
→ x+y-2√xy >0
→ x-2√xy+y >0
→ (√x-√y)^2 >0
In what conditions this inequality cannot hold?
Obviously, in case x is equal to y.

1 and 2 both shows that x is NOT equal to y, hence sufficient to tell that x+y>2√xy is true.

Q.E.D.

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gmatophobia · Answer

x+y>2\sqrt{xy}

x+y - 2\sqrt{xy} > 0

(\sqrt{x} - \sqrt{y})^2 > 0

Note:  (\sqrt{x} - \sqrt{y})^2  is always   non negative  , so if we are able to find out that   \sqrt{x}
eq{\sqrt{y}} , the value,  (\sqrt{x} - \sqrt{y})^2 , will always be positive and the answer to the question is yes.

If  \sqrt{x} = \sqrt{y} , the value  (\sqrt{x} - \sqrt{y})^2  is 0, and the answer to the question is No.

Statement 1

x > y

This means that x and y do not hold the same value, hence  \sqrt{x}
eq{\sqrt{y}}

This statement is sufficient, and we can rule out B, C and E

Statement 2

(x-y)^2>0

As  (x-y)^2>0 , we can conclude that the values of x and y are not same and that  \sqrt{x}
eq{\sqrt{y}} .

This statement is sufficient.

Option D

SaquibHGMATWhiz · Answer

Solution:      
  Pre Analysis:  
  We have  x+y>2\sqrt{xy}  
 Upon squaring both sides, we get  x^2+y^2+2xy>4xy 
 ⇒x^2+y^2-2xy>0 
 ⇒(x-y)^2>0  
 Generally speaking,  (x-y)^2\ge 0  but to be  (x-y)^2>0 ,  x and y has to be distinct

Statement 1:    x>y 
  From this, we get that x≠y and we can thus be sure that  (x-y)^2>0  
 Thus,  statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2:    (x-y)^2>0 
  This statement is directly telling us what we need to know and is thus sufficient

Hence the right answer is    Option D

If x and y are positive, is x + y> 2xy

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