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I apologize for the title, I couldn't figure out how to properly phrase the property of the constant e shown below:

$$10\ split\ in\ 1\ part\ multiplied\ by\ itself ={(10^1)} = 10$$ $$10\ split\ in\ 2\ parts\ multiplied\ by\ themselves ={(5^2)} = 25 \rightarrow value\ increases\ as\ we\ progress$$ $$10\ split\ in\ 3\ parts\ multiplied\ by\ themselves ={(3 \frac 13)^3)} \approx 37.03$$ $$10\ split\ in\ 10e^{-1}\ parts\ multiplied\ by\ themselves ={(e^{3.67})} \approx 39.3 = maximum\ value\ possible$$ $$10\ split\ in\ 4\ parts\ multiplied\ by\ themselves ={(2.5^4)} \approx 39.06$$ $$10\ split\ in\ 5\ parts\ multiplied\ by\ themselves ={(2^5)} = 32\ \rightarrow value\ decreases\ as\ we\ progress$$

This holds true for all numbers.

Apart from an intuitive understanding, I was wondering how this can be proved mathematically, or how we arrive at this conclusion logically, some method apart from visualization.

I have studied calculus and understood the derivation of e.

Thank you!

TheGamechanger
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    It looks the same issue as the one considered in this answer by Benjamin Dickman to a question of mine some time ago. – Jean Marie Jul 04 '22 at 16:44
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    You seem to be looking at $f(x)= \left(\frac{k}{x}\right)^x$ for $k=10$ and asking whether and why this is maximised over positive $x$ when $x=\frac{k}{e}$. I suggest you look at its derivative or at the derivative of $g(x)=\log(f(x)) = x \log(k) - x\log(x)$ – Henry Jul 04 '22 at 16:45
  • @Henry You should transform your comment into an answer. – Jean Marie Jul 04 '22 at 16:46
  • @Henry I understand it very well. I have observed that too... – Jean Marie Jul 04 '22 at 18:59

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