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\newtheorem{lemma}[theorem]{Lemma}
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\begin{document}

\title{Simple Proof That \( \pi ^ {3} < 3^{\pi} \)}
\author{Minco}
\date{\today}
\maketitle

\section*{Introduction}
This document presents a very simple mathematical proof that $\pi^3 < 3^{\pi}$.
The proof illustrates the beauty of pure mathematics, which allows a simple question to be shown in verbose manner.

\begin{theorem}
\[
  \pi^3 < 3^{\pi}
\]
\end{theorem}

\begin{proposition}
Since the both bases are over 1, we can exponentiate both sides of the inequality by \(\frac{1}{3 \pi}\).
So we have to prove that:
\[
  \pi^{\frac{1}{\pi}} < 3^{\frac{1}{3}}
\]
\end{proposition}

\section*{Limit Analysis}
Consider the function
\[
  f(x) = x^{\frac{1}{x}}, \quad x > 0.
\]

First, take the logarithm on both sides.

\[
  \log {f(x)} = \frac{1}{x} \log{x} \quad (\text{log is a natural logarithm})
\]
\[
  \therefore f(x) = e^{\frac{1}{x} \log{x}}
\]

Then analyze the limits at the boundaries of the domain:

\[
  \lim_{x \to 0^+} f(x) = \lim_{X \to - \infty} e^{X} = 0
\]

And

\[
  \lim_{x \to \infty} f(x) = \lim_{X \to 0^+} e^{X} = 1 .
\]

\section*{Derivative Analysis}

To find the derivative of \(f(x)\), use logarithmic differentiation as in Limit Analysis:

\[
y = x^{\frac{1}{x}} \implies \log y = \frac{\ln x}{x}.
\]

Differentiating both sides:
\[
\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\log x}{x}\right) = \frac{1 - \log x}{x^2}.
\]

Since \(x > 0\),

\[
  \frac{1- \log x}{x^2} = 0 \quad \rm{iff.} \quad 1 - \log x = 0
\]

\[
  \therefore x = e \quad \text{where} \quad \frac{d}{dx} f(x) = 0
\]

\section*{Graph}
\begin{center}
\begin{tikzpicture}
  \begin{axis}[
    domain=0.1:5, 
    samples=200,
    axis lines=middle,
    xlabel=$x$, ylabel={$f(x)$},
    ymin=0, ymax=2,
    ]
    \addplot [black, thick] {x^(1/x)};
    \addplot[
      only marks,
      mark=*,
      mark size=2pt,
      black
    ] coordinates {(3, {3^(1/3)})};
    \draw[dotted] (axis cs:3,0) -- (axis cs:3,{3^(1/3)});
    \draw[dotted] (axis cs:0,{3^(1/3)}) -- (axis cs:3,{3^(1/3)});
    \node [anchor=south east] at (axis cs:3,{3^(1/3)}) {$\bigl(3,\,3^{\frac{1}{3}}\bigr)$};
    
    \addplot[
      only marks,
      mark=*,
      mark size=2pt,
      black
    ] coordinates {(3.14, {3.14^(1/3.14)})};
    \draw[dotted] (axis cs:3.14,0) -- (axis cs:3.14,{3.14^(1/3.14)});
    \draw[dotted] (axis cs:0,{3.14^(1/3.14)}) -- (axis cs:3.14,{3.14^(1/3.14)});
    \node [anchor=south west] at (axis cs:3.14,{3.14^(1/3.14)}) {$\bigl(\pi,\,\pi^{\frac{1}{\pi}}\bigr)$};
  \end{axis}
\end{tikzpicture}
\end{center}

\begin{proof}
  As shown in the graph,

  \[
    \pi^{\frac{1}{\pi}} < 3^{\frac{1}{3}}
  \]
  \[
    \therefore \boxed{\pi^3 < 3^{\pi}}
  \]
\end{proof}
\end{document}