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

[Guest]

How can i prove that e^pi or pi^e is larger using calculus and no calculator? (remember: e is a constant)[/b]

 

[arthur ohlsten]

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

 

[Matt]

Hi calclover,

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:

Multiply both sides by e*pi to obtain:

and then:

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