Question

How to solve questions of this type??

"which is greater; x^y or y^x?"

Example- which is greater, 37^33 or 33^37 ?

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Answer 1

Answer:

Obviously 37^33 is only greater than 33^37 becoz 37 is greater than 33...

Therefore, x^y is greater...

Hope it helps...

Answer 2

Answer:

If they are positive integers, the answer is easy:

I will assume that x<y.

If 3≤x then xy>yx.

If x=2 then 2y<y2 for y=3; 2y=y2 for y=4; 2y>y2 for y>4.

If x and y are reals which exceed 1, then, since x1/x has a maximum at x=e and is increasing before and increasing after, if x≥e then xy>yx.

If x<e<y, interesting things can happen. I recall a result of mine from probably 40 or more years ago of the form "if 1<x<e<y and xy>e2 then xy<yx". The last inequality may go the other way. I'll see if I can find or prove it