 |
| ||
| |||
|
| Math Central | Quandaries & Queries |
|
Question from natalie, a student: Without using your calculator, prove which is bigger: e^pi or pi^e we talked about ad absurdum in class, so I'm assuming that is how I approach this question, but since neither of them have a variable( like the examples we actually solved in class), I'm not quite sure how to solve it.... |
Natalie,
Look at the last paragraph on
mathcentral.uregina.ca/QQ/database/QQ.09.00/dusty1.html
and see if that helps. If you need more assistance write back.
Harley
Natalie wrote back
Hi,
I'm still confused. How does creating a (lnx)/x help us get to the conclusion? I'm not even sure why it is a graph of (lnx)/x when it is about e^pi and pi^e??
thank you,
natalie
Natalie,
If f(x) = ln(x)/x and the maximum value of f(x) occurs at x = e the f(e) > f(π). Thus
ln(e)/e > ln(π)/π.
But ln(e) = 1 so
1/e > ln(π)/π.
Can you manipulate this inequality to show that eπ > πe? If so then all that remains is to show that the maximum value of f(x) occurs at x = e. I used calculus.
Harley
| ||
| ||
| ||
 |
 |
|
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.