Which is greater: $e^{\pi}$ or $\pi^{e}$?

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SUMMARY

The discussion centers on the comparison of the values of $e^{\pi}$ and $\pi^{e}$. It concludes that $e^{\pi}$ is greater than $\pi^{e}$ by employing calculus inequalities. The method involves raising both expressions to the power of $1/(e\pi)$ and analyzing the maximum of the function $x^{1/x}$. This approach provides a definitive mathematical framework for the comparison.

PREREQUISITES
  • Understanding of calculus inequalities
  • Familiarity with exponential functions
  • Knowledge of the mathematical constants $e$ and $\pi$
  • Basic skills in function analysis and optimization
NEXT STEPS
  • Study the properties of exponential functions and their graphs
  • Learn about calculus inequalities and their applications
  • Explore the function $x^{1/x}$ and its maximum value
  • Investigate other comparisons involving mathematical constants
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Mathematicians, calculus students, and anyone interested in exploring mathematical inequalities and the properties of exponential functions.

Poly1
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Which is greater, $e^{\pi}$ or $\pi^{e}$?

I found this when searching for calculus inequalities.
 
Last edited by a moderator:
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A variation of the same method is to raise both numbers to power $1/(e\pi)$ and find the maximum of $x^{1/x}$.
 

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