PLEASE HELP ASAP!

G

Guest

Guest
I wrote:
How can i prove that e^pi or pi^e is larger using calculus and no calculator? (remember: e is a constant)

This is the response I got:
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:

****MY 2ND QUESTION: how did you come up with f(x)=lnx/x?


Multiply both sides by e*pi to obtain:



and then:



since e^x is a strictly increasing function. Thus, pi^e < e^pi.
 
the proof you received was elegant, but here is a approach that you might find easier
is e^pi > pi^e ?

let us replace pi with x , to help us see the problem easier, then we will let x be pi. this is not neccessary but helps me understand the problem.

is e^x > x^e ?
let us plot e^x and x^e for 0<x
table of values
x e^x x^e
0 1 0
1 e 1
2 e^2 2^e
3 e^3 3^e

where do the two curves cross? e^x= x^e take ln
x = e lnx or
x-e lnx =0 solve by newtons method or any other method you have studied x is slightly less than 2

so for x<2 e^x>x^e
for x>2 x^e>e^x let x=pi which is greater than 2
pi^e > e^pi
Arthur
 
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