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

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The discussion centers on comparing the values of $e^{\pi}$ and $\pi^{e}$. A method suggested involves raising both expressions to the power of $1/(e\pi)$ to analyze their growth rates. This approach relates to finding the maximum of the function $x^{1/x}$. The conversation highlights the mathematical exploration of inequalities in calculus. Ultimately, the comparison seeks to determine which of the two expressions is greater.
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Which is greater, $e^{\pi}$ or $\pi^{e}$?

I found this when searching for calculus inequalities.
 
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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 $x^{1/x}$.
 
Thread 'Problem with calculating projections of curl using rotation of contour'

Thread 'Problem with calculating projections of curl using rotation of contour'

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