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+\documentclass[12pt]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath, amssymb, amsthm}
+\usepackage{pgfplots}
+\usepackage{geometry}
+\geometry{margin=1in}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+
+\theoremstyle{remark}
+\newtheorem*{remark}{Remark}
+
+% Document
+\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}