is e^pi bigger or is pi^e bigger?

G

Guest

Guest
How can i prove that e^pi or pi^e is larger using calculus and no calculator? (remember: e is a constant)[/b]
 
e^pi >? pi^e
e^x>x^e x a variable greater than 0 find range of x
take derivative with respect to x
e^x>e x>1
then for all values of x greater than 1 e^x is greater than x^e
then for x=pi e^pi is greater than pi^e QED

please check with a calculator to check on the answer I have no calculator
Arthur
 
Hi calclover,

calclover said:
How can i prove that e^pi or pi^e is larger using calculus and no calculator? (remember: e is a constant)
Let f(x) = ln(x)/x. First show that f'(x) < 0 for all x > e (which means that f is a strictly decreasing function for x ≥ e). Then:

pie1.png


Multiply both sides by e*pi to obtain:

pie2.png


and then:

pie3.png


since e^x is a strictly increasing function. Thus, pi^e < e^pi.
 
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