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Question from natalie, a student:

Without using your calculator, prove which is bigger: e^pi or pi^e

we talked about ad absurdum in class, so I'm assuming that is how I approach this question, but since neither of them have a variable( like the examples we actually solved in class), I'm not quite sure how to solve it....

Natalie,

Look at the last paragraph on
http://mathcentral.uregina.ca/QQ/database/QQ.09.00/dusty1.html
and see if that helps. If you need more assistance write back.

Harley

Natalie wrote back

Hi,

I'm still confused. How does creating a (lnx)/x help us get to the conclusion? I'm not even sure why it is a graph of (lnx)/x when it is about e^pi and pi^e??

thank you,
natalie

Natalie,

If f(x) = ln(x)/x and the maximum value of f(x) occurs at x = e the f(e) > f(π). Thus

ln(e)/e > ln(π)/π.

But ln(e) = 1 so

1/e > ln(π)/π.

Can you manipulate this inequality to show that eπ > πe? If so then all that remains is to show that the maximum value of f(x) occurs at x = e. I used calculus.

Harley

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