[Tex/LaTex] Intersection of lines to draw a convex function

Question

I'm trying to draw (in TikZ) a convex function f constructed as the minimum of the linear functions
f₁,
f₂,
f₃ and
f₄.
To do that, my inclination is to use the intersections of said lines to find the minimum, but the line intersection doesn't seem to allow me to do that.
Concretely, I have the following code:

\documentclass{article}
\nonstopmode

\usepackage{tikz}

\usetikzlibrary{calc,intersections}

\begin{document}
\begin{tikzpicture}[scale=.5]

    \draw[-stealth] (0,0) -- (10,0) node [below right] {};
    \draw[-stealth] (0,0) -- (0,10) node [left=2] {};

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \foreach \a/\b [count=\i] in {60/1.7, 45/2.4, 30/3.7, 18/5.5} {
        \begin{scope}
            \clip (0,0) rectangle (10,10);
            \draw (0,\b) -- +(\a:20);
        \end{scope}
        \node [left] at (0,\b) {$f_\i$};
    }

    % the following doesn't work because the "intersection of" doesn't like
    % the coordinates, or so it seems
    %\def\lasta{90}\def\lastb{0}
    %\draw[red, thick] (0,0) 
    %    \foreach \a/\b [remember=\a as \lasta, remember=\b as \lastb]
    %        in {60/1.7, 45/2.4, 30/4, 23/5} -- 
    %        (intersection of (0,\lastb)--+(\lasta:20) and (0,\b)--+(\a,20));
\end{tikzpicture}
\end{document}

I'd like to draw, with a fat red line, the function f that is the point-wise minimum of all of those, in a generic way.
Any comments are welcome, but my question boils down to this:

How can I take the intersections of said lines, without manually unfolding the loops?

Answer 1

With only TikZ but with functions. Remark : I use samples=2 to draw the lines because two points are enough but for m we need enough points to find the correct min values. I use a part of the Jake's answer.

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[declare function={
    f(\t)=tan(60)*\t+1.7;
    g(\t)=tan(45)*\t+2.4;
    h(\t)=tan(30)*\t+3.7;
    i(\t)=tan(18)*\t+5.5;
    m(\t)=min(f(\t) ,g(\t) ,h(\t),i(\t));}] 

 \draw[very thin,color=gray] (-0.1,-0.1) grid (10.1,10);
 \draw[->] (-0.2,0) -- (10.2,0) node[right] {$x$};
 \draw[->] (0,-0.2) -- (0,10.2) node[above] {$y$};  

 \clip (-1,-1) rectangle (10,10);     
\foreach \func in {f,g,h,i}
\draw [blue, thin] plot [domain=0:10, samples=2] (\x,{\func(\x)});
\draw [red, thick] plot [domain=0:10, samples=100] (\x,{m(\x)}); 
\end{tikzpicture}

\end{document}