142 lines
3.1 KiB
TeX
142 lines
3.1 KiB
TeX
\documentclass[12pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath, amssymb, amsthm}
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\usepackage{pgfplots}
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\usepackage{geometry}
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\geometry{margin=1in}
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\newtheorem{theorem}{Theorem}
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{corollary}[theorem]{Corollary}
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\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{example}[theorem]{Example}
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\theoremstyle{remark}
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\newtheorem*{remark}{Remark}
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% Document
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\begin{document}
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\title{Simple Proof That \( \pi ^ {3} < 3^{\pi} \)}
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\author{Minco}
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\date{\today}
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\maketitle
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\section*{Introduction}
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This document presents a very simple mathematical proof that $\pi^3 < 3^{\pi}$.
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The proof illustrates the beauty of pure mathematics, which allows a simple question to be shown in verbose manner.
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\begin{theorem}
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\[
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\pi^3 < 3^{\pi}
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\]
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\end{theorem}
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\begin{proposition}
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Since the both bases are over 1, we can exponentiate both sides of the inequality by \(\frac{1}{3 \pi}\).
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So we have to prove that:
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\[
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\pi^{\frac{1}{\pi}} < 3^{\frac{1}{3}}
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\]
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\end{proposition}
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\section*{Limit Analysis}
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Consider the function
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\[
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f(x) = x^{\frac{1}{x}}, \quad x > 0.
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\]
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First, take the logarithm on both sides.
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\[
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\log {f(x)} = \frac{1}{x} \log{x} \quad (\text{log is a natural logarithm})
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\]
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\[
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\therefore f(x) = e^{\frac{1}{x} \log{x}}
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\]
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Then analyze the limits at the boundaries of the domain:
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\[
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\lim_{x \to 0^+} f(x) = \lim_{X \to - \infty} e^{X} = 0
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\]
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And
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\[
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\lim_{x \to \infty} f(x) = \lim_{X \to 0^+} e^{X} = 1 .
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\]
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\section*{Derivative Analysis}
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To find the derivative of \(f(x)\), use logarithmic differentiation as in Limit Analysis:
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\[
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y = x^{\frac{1}{x}} \implies \log y = \frac{\ln x}{x}.
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\]
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Differentiating both sides:
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\[
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\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\log x}{x}\right) = \frac{1 - \log x}{x^2}.
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\]
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Since \(x > 0\),
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\[
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\frac{1- \log x}{x^2} = 0 \quad \rm{iff.} \quad 1 - \log x = 0
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\]
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\[
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\therefore x = e \quad \text{where} \quad \frac{d}{dx} f(x) = 0
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\]
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\section*{Graph}
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\begin{center}
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\begin{tikzpicture}
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\begin{axis}[
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domain=0.1:5,
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samples=200,
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axis lines=middle,
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xlabel=$x$, ylabel={$f(x)$},
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ymin=0, ymax=2,
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]
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\addplot [black, thick] {x^(1/x)};
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\addplot[
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only marks,
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mark=*,
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mark size=2pt,
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black
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] coordinates {(3, {3^(1/3)})};
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\draw[dotted] (axis cs:3,0) -- (axis cs:3,{3^(1/3)});
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\draw[dotted] (axis cs:0,{3^(1/3)}) -- (axis cs:3,{3^(1/3)});
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\node [anchor=south east] at (axis cs:3,{3^(1/3)}) {$\bigl(3,\,3^{\frac{1}{3}}\bigr)$};
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\addplot[
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only marks,
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mark=*,
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mark size=2pt,
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black
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] coordinates {(3.14, {3.14^(1/3.14)})};
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\draw[dotted] (axis cs:3.14,0) -- (axis cs:3.14,{3.14^(1/3.14)});
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\draw[dotted] (axis cs:0,{3.14^(1/3.14)}) -- (axis cs:3.14,{3.14^(1/3.14)});
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\node [anchor=south west] at (axis cs:3.14,{3.14^(1/3.14)}) {$\bigl(\pi,\,\pi^{\frac{1}{\pi}}\bigr)$};
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\end{axis}
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\end{tikzpicture}
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\end{center}
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\begin{proof}
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As shown in the graph,
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\[
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\pi^{\frac{1}{\pi}} < 3^{\frac{1}{3}}
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\]
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\[
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\therefore \boxed{\pi^3 < 3^{\pi}}
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\]
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\end{proof}
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\end{document}
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