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\begin{tcolorbox}[breakable, colframe=white!10!jingga, coltitle=white!90!jingga, colback=white!95!jingga, coltext=black!90!brown!80!jingga, colbacktitle=white!10!jingga, enhanced, fonttitle=\bfseries,fontupper=\normalsize, attach boxed title to top left={yshift=-2mm}, before skip=8pt, after skip=8pt, title=2024-03-12]
已知 $ f(x) $ 在区间 $ [0,2] $ 上具有三阶连续导数，设$$g(x)=f(0)+(f(1)-f(0)) x+\frac{f(2)-2 f(1)+f(0)}{2}\left(x^{2}-x\right)$$试证明: $ \max\limits _{x \in[0,2]}|f(x)-g(x)| \leq \dfrac{\sqrt{3}}{27} \max\limits _{x \in[0,2]}\left|f^{\prime \prime \prime}(x)\right| $. \tcblower
对 $ g(x) $ 变形得：$$\begin{aligned}g(x) & =f(0)+\frac{f(1)-f(0)}{1-0} \cdot x+\frac{f(2)-f(1)-[f(1)-f(0)]}{2-0} \cdot x \cdot(x-1) \\& =f[0]+f[0,1] x+f[0,1,2] \cdot x \cdot(x-1) .\end{aligned}$$易知 $ g(x) $ 为二次牛顿插值多项式. 因为 $ f(x) $ 为定义在 $ [0,2] $ 上且具有三阶连续导数的函数， $ x_{0}=0 , x_{1}=1 , x_{2}=2 $ 是 $[0.2]$上的 $3$ 个互异插值节点,则 $$R_{2}(x)=f(x)-g(x)=f[0,1,2,x] \cdot(x-0)(x-1)(x-2)$$而又知 $ f[0,1,2，x]=\dfrac{f^{(3)}(\xi)}{3 !} $,则 $$ \max _{x \in[0,2]}|f(x)-g(x)|=\max _{x \in[0,2]}\left|\frac{f^{(3)}(\xi)}{6} \cdot x(x-1)(x-2)\right| $$令$h(x)=x \cdot(x-1)(x-2)=x^{3}-3 x^{2}+2 x,x \in[0,2]$.由$ h^{\prime}(x)=3 x^{2}-6 x+2=0 $得$ x_{1}=1-\frac{\sqrt{3}}{3}, x_{2}=1+\frac{\sqrt{3}}{3} $.计算得到$$h(0)=0, \quad h(2)=0 . \quad h\left(x_{1}\right)=\frac{2 \sqrt{3}}{9} \quad h\left(x_{2}\right)=-\frac{2 \sqrt{3}}{2}$$于是 $ \max\limits _{x \in[0, 2]}|h(x)|=\frac{2}{9} \sqrt{3} $.
$$\begin{aligned}\max _{x \in[0,2]}|f(x)-g(x)|&=\max _{x \in[0,2]}\left|\frac{f^{\prime \prime \prime}(\xi)}{6} \cdot x(x-1)(x-2)\right| \\&\leqslant \frac{1}{6} \max _{x \in[0,2]}\left|f^{\prime \prime \prime}(x)\right| \cdot \max _{x \in[0,2]}\left|\left(x-{0}\right)\left(x-{1}\right)\left(x-{2}\right)\right| \\&=\frac{1}{6} \cdot\frac{2 \sqrt{3}}{9} \max _{x \in[0,2]}\left|f^{\prime \prime \prime}(x)\right|\\&=\frac{\sqrt{3}}{27} \max _{x \in[0,2]}\left|f^{\prime \prime \prime}(x)\right|\end{aligned}$$得证.\end{tcolorbox}
main.tex 文件设定
%%%%%\documentclass{article}\usepackage[paperwidth=210mm, paperheight=297mm, margin=2cm]{geometry}%\usepackage[utf8]{inputenc}%\usepackage{ebgaramond-maths}\usepackage{ctex}%==Math & Physics Packages==\usepackage{amsmath, amsthm, amsfonts, amssymb}\usepackage{setspace}\usepackage{physics}\usepackage{cancel}\usepackage{nicefrac}
%==Image-related Packages==\usepackage{graphics, graphicx}\usepackage{tikz}\usetikzlibrary{arrows.meta}\usepackage{pgfplots}\pgfplotsset{compat=1.18}\usepackage{fourier-orns}\usepackage{lipsum}%==Colour Palette==\definecolor{merah}{HTML}{F4564E}\definecolor{merahtua}{HTML}{89313E}\definecolor{biru}{HTML}{60BBE5}\definecolor{birutua}{HTML}{412F66}\definecolor{hijau}{HTML}{59CC78}\definecolor{hijautua}{HTML}{366D5B}\definecolor{kuning}{HTML}{FFD56B}\definecolor{jingga}{HTML}{FBA15F}\definecolor{ungu}{HTML}{8C5FBF}\definecolor{lavender}{HTML}{CBA5E8}\definecolor{merjamb}{HTML}{FFB6E0}
%==Theorems==\usepackage{xcolor}\usepackage{tcolorbox}\tcbuselibrary{skins,breakable,theorems}\usepackage{changepage}%用于调整页面布局，例如改变页面的宽度或高度。\renewcommand{\thesection}{\arabic{section}.}
\makeatletter
% Definition% \newtcbtheorem{dfn}{1}% {breakable,enhanced,arc=0mm,outer arc=0mm,% boxrule=0pt,toprule=1pt,leftrule=0pt,bottomrule=1pt, rightrule=0pt,left=0.2cm,right=0.2cm, % titlerule=0.5em,toptitle=0.1cm,bottomtitle=-0.1cm,top=0.2cm, % colframe=white!10!biru,colback=white!90!biru,coltitle=white, % coltext=birutua!60!gray, title style={white!10!biru}, before skip=8pt, after skip=8pt, %fonttitle=\bfseries,fontupper=\normalsize}{dfn} %==Formatting Packages==\usepackage{multicol}\usepackage{multirow}
\usepackage{enumitem}\usepackage{indentfirst}\usepackage[toc]{multitoc}
%==Layouting Packages==\usepackage{titlesec}\usepackage{fancyhdr}\pagestyle{fancy}\setlength{\headheight}{15pt}% Uncomment jika perlu\fancyhead[L]{Numerical Analysis}\fancyhead[R]{xiaowen}%\fancyfoot[L]{}\fancyfoot[C]{\thepage}%\fancyfoot[R]{\thepage}
\renewcommand\headrule{%定义页眉中的水平线样式的 \vspace{-1pt}%-1pt表示向上调整1个点的垂直间距 \hrulefill \raisebox{-2.1pt} \hrulefill}
\renewcommand\footrule{% \hrulefill \raisebox{-2.1pt} %{\quad\floweroneleft\decoone\floweroneright\quad}% \hrulefill} %Reference and Bibliography Packages==设置超链接的样式和颜色\usepackage{hyperref}\hypersetup{ colorlinks=true, linkcolor={black}, citecolor={biru!70!black}, urlcolor={biru!80!black}}
代码参考：https://www./index/details/index/mid/3762.html
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