Prove: e^(x+y)/(x*y)>=e^2      if x,y>0

Expert Answers

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We have to prove that

`e^(x+y)/(xy)gt= e^2`

Lets take the function y=x-ln(x)

`y' = 1 - 1/x` y'=0 when x=1

This has a minimum at x=1  (we could use the 2nd derivative test to prove this is a minimum).

It's value is 1-ln(1)=1  So x-ln(x) is a function with an absolute minimum at x=1 and it's value is 1, so x-ln(x)>=1

x-ln(x)>=1 and y-ln(x)>=1

x-ln(x) + y-ln(y) >= 2

x+y - ln(xy) >= 2

`e^(x+y)/e^(ln(xy)) gt= e^2`

`e^(x+y)/(xy) gt= e^2`

And we have `e^(x+y)/xy gt= e^2`

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