如何自动绘制拐点处的切线？

Question

功能

f(x) = \frac{x^4}{12}-\frac{2x^3}{3}+\frac{3 x^2}{2}

有两个拐点x = 1和x=3。我手工绘制了上述点处的切线。

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usepgfplotslibrary{fillbetween}
\begin{document}
    \begin{tikzpicture}[>=latex]
    \begin{axis}[
    xtick={-2,-1,...,5},
    xticklabel=\empty, 
    ytick={-2,-1,...,5},
    yticklabel=\empty, grid=major,
    axis x line=center,
    axis y line=center,
    xlabel={$x$},
    ylabel={$y$},
    xlabel style={below},
    ylabel style={left},
    xmin=-2,
    xmax=5,
    ymin=-2,
    ymax=5]
        \addplot[color=blue,smooth,samples=501, thick,domain=-5:5] {(3* x^2)/2 - (2 *x^3)/3 + x^4/12};
    \addplot[color=purple,smooth,samples=501, thick,domain=-5:5] {-(5/12) + (4 *x)/3};
    \addplot[color=red,smooth,samples=501, thick,domain=-5:5] {9/4};
    \end{axis}
    \end{tikzpicture}
\end{document}

如何自动绘制每个函数拐点处的切线

Answer 1

这只是为了好玩。它确实以合理的精度在拐点处产生了切线。Ti钾Z 根据输入函数计算它们的位置。

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{calc}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usepgfplotslibrary{fillbetween}
    \makeatletter
\def\parsenode[#1]#2\pgf@nil{%
    \tikzset{label node/.style={#1}}
    \def\nodetext{#2}
}

\tikzset{
    add node at x/.style 2 args={
        name path global=plot line,
        /pgfplots/execute at end plot visualization/.append={
                \begingroup
                \@ifnextchar[{\parsenode}{\parsenode[]}#2\pgf@nil
            \path [xshift=-0.1pt,name path global = position line #1-1]
                ({axis cs:#1,0}|-{rel axis cs:0,0}) --
                ({axis cs:#1,0}|-{rel axis cs:0,1});
            \path [xshift=0.1pt, name path global = position line #1-2]
                ({axis cs:#1,0}|-{rel axis cs:0,0}) --
                ({axis cs:#1,0}|-{rel axis cs:0,1});
            \path [
                name intersections={
                    of={plot line and position line #1-1},
                    name=left intersection
                },
                name intersections={
                    of={plot line and position line #1-2},
                    name=right intersection
                },
                label node/.append style={pos=1}
            ] (left intersection-1) -- (right intersection-1)
            node [label node]{\nodetext};
            \endgroup
        }
    }
}
\pgfplotsset{tangent/.style={ 
add node at x={#1}{ 
[ 
sloped, 
append after command={(\tikzlastnode.west) edge [thick,black] (\tikzlastnode.east)}, 
minimum width=0.2\textwidth 
] 
} 
}}
\makeatother 
\begin{document}
    \begin{tikzpicture}[>=latex,declare function={f(\x)=(3* pow(\x,2))/2 - 
    (2 *pow(\x,3))/3 + pow(\x,4)/12;difff(\x)=(f(\x+0.02)-f(\x-0.02))/0.04;
    ddifff(\x)=(difff(\x+0.02)-difff(\x-0.02))/0.04;}]
    \begin{scope}[opacity=0,overlay]
     \draw[name path=ddiff] plot[variable=\x,domain=-2:5] (\x,{ddifff(\x)});
     \draw[name path=xaxis] (-2,0) -- (5,0);
     \path[name intersections={of={ddiff and xaxis},total=\t}]
     foreach \X in {1,...,\t} {let \p1=(intersection-\X) in
     \pgfextra{\pgfmathparse{\x1/1cm}
     \ifnum\X=1
     \xdef\LstInfPts{\pgfmathresult}
     \else
     \xdef\LstInfPts{\LstInfPts,\pgfmathresult}
     \fi}};
    \end{scope} 
    \begin{axis}[
    xtick={-2,-1,...,5},
    xticklabel=\empty, 
    ytick={-2,-1,...,5},
    yticklabel=\empty, grid=major,
    axis x line=center,
    axis y line=center,
    xlabel={$x$},
    ylabel={$y$},
    xlabel style={below},
    ylabel style={left},
    xmin=-2,
    xmax=5,
    ymin=-2,
    ymax=5]
    \addplot[color=blue,smooth,samples=501, thick,domain=-5:5,
    tangent/.list/.expanded=\LstInfPts] {f(x)};
    \end{axis}
    \end{tikzpicture}
\end{document}