# Determining the bigger number

## Homework Statement

Use the function ##f(x)=x^{1/x} \, ,\, x>0##, to determine the bigger of two numbers ##e^{\pi}## and ##\pi^e##.

## The Attempt at a Solution

I honestly don't know where to begin with this problem. I found the derivative but that seems to be of no help. The function increases when x>1/e and decreases when 0<x<1/e.

Any help is appreciated. Thanks!

mfb
Mentor
Hint: a<b is equivalent to a^c < b^c for positive a,b,c. Can you find a c such that your expressions look like x^(1/x)?

haruspex
Homework Helper
Gold Member
2020 Award
. The function increases when x>1/e and decreases when 0<x<1/e.
Doesn't sound right. What do you get for x = 1, 2, 4, 16?

1 person
Doesn't sound right. What do you get for x = 1, 2, 4, 16?

Oops, its just the opposite of what I posted.

I have tried the problem again.
##f(e)>f(\pi) \Rightarrow e^{1/e}>\pi^{1/\pi}##
Raising both the sides to the power ##e\pi##
$$e^{\pi}>\pi^e$$

Looks correct?

Dick
Homework Helper
Oops, its just the opposite of what I posted.

I have tried the problem again.
##f(e)>f(\pi) \Rightarrow e^{1/e}>\pi^{1/\pi}##
Raising both the sides to the power ##e\pi##
$$e^{\pi}>\pi^e$$

Looks correct?

Looks correct.

Looks correct.

Thank you Dick and haruspex!