Second Derivative

I modify my answer to get what you want ...

My answer is not complete because I need more time ...

My package is useful for french mathematics teachers. We never make tables like you want. Below I made the "traditional" table ...

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{tkz-tab,tkz-fct}
\usetikzlibrary{arrows}

\begin{document}

\noindent\begin{tikzpicture}[xscale=1]
    \tkzTabInit[espcl=1.5]{$x$/1,$f’(x)$/2,$f’'(x)$/2,$f(x)$/4} 
 {$-\infty$,$0$,$2$,$3$,$+\infty$}%
\tkzTabLine{,-,0,-, ,-,0,+, }%
\tkzTabLine{,+,0,-,0,+, ,+, }%
\tkzTabVar{+/ $+\infty$ ,R/ ,R/ ,-/-16,+/ $+\infty$ } % french method

\end{tikzpicture}
\begin{tikzpicture}
\tkzInit[xmin=-1,xmax=4,xstep=1,
        ymin=-16,ymax=16,ystep=8]
\tkzAxeXY
\tkzFct[color=red,samples=100,domain = -1:4]{\x**4-4*\x**3+11}
\end{tikzpicture}
\end{document}

With the french method, it's usual to place some intermediate values like 11, -5 or -16

If you want to personalize the result you can use the help option. With this option you get

Now you have all the nodes defined in the picture. With this coordinates you can place what you want.

Now you can use N13 , N14 etc. to place the nodes and to draw some curve arrows. I can post a complete answer tomorrow but perhaps you can find the solution ...

Final

Now we know the nodes necessary for placing the curved arrows. We can place the values like this

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tkz-tab}
\usetikzlibrary{arrows}

\begin{document}

\begin{tikzpicture}
\tkzTabInit[espcl=1.5]{$x$/1,$f’(x)$/2,$f’'(x)$/2,$f(x)$/4} {$-\infty$,$0$,$2$,$3$,$+\infty$}%
\tkzTabLine{,-,0,-, ,-,0,+, }%
\tkzTabLine{,+,0,-,0,+, ,+, }%
\node [below] at (N13){\scriptsize$+\infty$};
\node [below=1cm] at (N23){\scriptsize$11$};
\node [below=2cm] at (N33){\scriptsize$-5$};
\node [above] at (N44){\scriptsize$-16$};
\node [below] at (N53){\scriptsize$+\infty$};
\end{tikzpicture}

\end{document}

Th next step is to draw arrows ... It's easy and there are several possibilities. We can give a name to the last nodes and ...

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{tkz-tab}
\usetikzlibrary{arrows}

\begin{document}

\begin{tikzpicture}

\tkzTabInit[espcl=2]{$x$/1,$f’(x)$/2,$f’'(x)$/2,$f(x)$/4} {$-\infty$,$0$,$2$,$3$,$+\infty$}
\tkzTabLine{,-,0,-, ,-,0,+, }
\tkzTabLine{,+,0,-,0,+, ,+, }

\node [below] (n1) at (N13){\scriptsize$+\infty$};
\node [below=1cm](n2) at (N23){\scriptsize$11$};
\node [below=2cm] (n3) at (N33){\scriptsize$-5$};
\node [above] (n4) at (N44){\scriptsize$-16$};
\node [below] (n5) at (N53){\scriptsize$+\infty$};

\draw[>->] (n1) to [out=-90,in=180] (n2);
\draw[>->] (n2) to [out=0,in=90] (n3);
\draw[>->] (n3) to  [out=-90,in=180] (n4);
\draw[>->] (n4) to [out=0,in=-90] (n5);

\end{tikzpicture}

\end{document}

Final remarks: I used kz-fct to draw the 2D plot but you can make this step independently. There iis no link finally with your question. You can draw this curve only with tikz or with the good tool PGFPlots. A lot of guys here can help you for this. tkz-fct needs to install gnuplot and sometimes it's not easy.

Here is a solution using PSTricks drawings inside a table. (Compile using latex --> dvips --> ps2pdf og xelatex.)

\documentclass{article}

\usepackage[
  tableposition = top,
  labelformat = empty % Removes ``Table'' or ``Figure'' from caption.
]{caption}
\usepackage{pstricks}
\usepackage{booktabs}

% Increasing function.
\newcommand*\increase{
  \raisebox{-1.2ex}{%
    \begin{pspicture}(0.5,0.5)
      \psline[linecolor = blue]{->}(0,0)(0.5,0.5)
    \end{pspicture}%
  }
}

% Decreasing function.
\newcommand*\decrease{
  \raisebox{-1.2ex}{%
    \begin{pspicture}(0.5,0.5)
      \psline[linecolor = blue]{->}(0,0.5)(0.5,0)
    \end{pspicture}%
  }
}

\begin{document}

\begin{table}
 \centering
  \caption{Monotony of a function~$f$.}
  \label{tbl:1}
  \renewcommand\arraystretch{1.5}
  \begin{tabular}{*{10}{c}}
    \toprule
     $x$      & $-\infty$ &           & $0$ &           & $1$ &           & $2$ &           & $+\infty$ \\
    \midrule
     $f'(x)$  &           & $+$       &     & $-$       &     & $-$       &     & $+$       &           \\
     $f''(x)$ &           & $-$       &     & $-$       &     & $+$       &     & $+$       &           \\
     $f(x)$   &           & \increase &     & \decrease &     & \decrease &     & \increase &           \\
    \bottomrule
   \end{tabular}
\end{table}

\end{document}

Note: I know it's not exactly what you asked for but I like this solution, so please consider it as my suggestion of how to do it.

This may help in putting curved arrows in place:

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{tikz,tkz-tab,amsmath}
\usetikzlibrary{arrows}

\begin{document}

\newcommand{\E}{\mathrm{e}}
\begin{tikzpicture}
\tikzset{arrow style/.style = {blue,->,> = latex',
shorten > = 6pt,
shorten < = 6pt}}
\tkzTabInit[espcl=2]{$x$ /1, $f'(x)$ /1.5, $f''(x)$ /2,$f(x)$/2.5}{$-\infty$, $0$ ,$1$ , $2$ , $+\infty$}%
\tkzTabLine{ ,+,z,- ,d,-,z,+}%
\tkzTabLine{ ,-,t ,- ,d,+,t ,+}%
% \tkzTabVar%
% {  -/  ,  +/ ,-D+/  /  ,  -/  ,+/  /  }%
\pgfnodebox{i1}[virtual]{\pgfxy(2.5,-5)}{$+\infty$}{0pt}{0pt}
\pgfnodebox{n1}[virtual]{\pgfxy(4.5,-5.5)}{$11$}{0pt}{0pt}
\pgfnodebox{n2}[virtual]{\pgfxy(6.5,-6)}{$-5$}{0pt}{0pt}
\pgfnodebox{n3}[virtual]{\pgfxy(8.5,-6.5)}{$-16$}{0pt}{0pt}
\pgfnodebox{i2}[virtual]{\pgfxy(10.5,-5)}{$+\infty$}{0pt}{0pt}
\pgfsetendarrow{\pgfarrowtriangle{4pt}}
\pgfnodeconncurve{i1}{n1}{-65}{-180}{.5cm}{.5cm}
\pgfnodeconncurve{n1}{n2}{0}{110}{.5cm}{.5cm}
\pgfnodeconncurve{n2}{n3}{-80}{-180}{.5cm}{.5cm}
\pgfnodeconncurve{n3}{i2}{0}{-90}{1.5cm}{1cm}
\end{tikzpicture}

\end{document}

I'm sure there is a more efficient way of producing the same kind of output: