e^pi vs. pi^e
Date: 03/20/2002 at 14:51:19 From: Rajesh Gunesh Subject: Number Theory Which is greater, e^pi or pi^e? I would like to have a simple proof, please. Thank you!
Date: 03/20/2002 at 15:28:22 From: Doctor Paul Subject: Re: Number Theory Consider the function f(x) = e^x - x^e. Its derivative is f'(x) = e^x - e*x^(e-1) f'(x) = 0 will have at most two solutions since f'(x) = 0 implies e^x = e*x^(e-1) and it is easily seen that an exponential function and a power function have at most two points of intersection. Observe that f'(1) = 0 and f'(e) = 0, so x = 1 and x = e are the two points of intersection. In general, an exponential function will grow to infinity faster than a power function, so f'(x) >= 0 for x > e. This means that f is an increasing function when x > e. Also notice that f(e) = 0. Thus, since Pi > e and f is increasing when x > e, we have f(Pi) > 0. Thus f(Pi) = e^Pi - Pi^e > 0. This implies that e^Pi is greater than Pi^e. A calculator verifies that e^Pi ~= 23.14 and Pi^e ~= 22.46 I hope this helps. Please write back if you'd like to talk about this some more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/
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