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\node [left] at (i1-|0,0) {$x^{\ast}_1$};

\documentclass{article}
\usepackage{tikz} 
\usepackage{amsmath} 
\usetikzlibrary{intersections,calc} 

\begin{document}


\centering 
\begin{tikzpicture}[domain=0:6, range=4:5, thick] 
\pgfmathsetlengthmacro{\lengthXaxis}{6cm} 
\pgfmathsetlengthmacro{\lengthYaxis}{5cm} 
\pgfmathsetlengthmacro{\radiusArcFigureOne}{5.5cm} 

\draw [->] (0,0) -- (0,\lengthYaxis) node[above] {$\pmb{x_1,\textcolor{cyan}{x_2}}$}; 
\draw [->] (0,0) -- (\lengthXaxis,0) node[right] {$\pmb{\textcolor{green}{y}}$}; 

\draw [name path global=f1a] (\radiusArcFigureOne,0) 
       arc [ 
          start angle=0, 
          end angle=90, 
          x radius=\radiusArcFigureOne, 
          y radius=(3*\radiusArcFigureOne/4)-2mm 
      ] 
; 
\draw[name path global=f1b, thick, color=purple, domain=2.5:8, shift={(-2,.7)}] 
       plot (\x,{25*exp(-.7*\x-.25)+.7}) node[right] {$\bar{u}_2$} 
; 

\begin{scope}[yshift=-6cm]  
\draw [->, name path global=f2xline] (0,0) -- (\lengthXaxis,0) node[right] {$\pmb{\textcolor{green}{y}}$}; 

\draw [->] (0,0) -- (0,\lengthYaxis) node[above] {$\pmb{x_1}$}; 

\draw[name path global=f5, thick, color=cyan, domain=4:8, shift={(-2,-.067)}] 
       plot (\x,{15*exp(-.5*\x-.35)+.75}) node[right] {$u^{\ast}_1$};
\end{scope} 


\begin{scope}[yshift=-12cm] 
\draw [->] (0,0) -- (0,\lengthYaxis) node[above] {$\textcolor{cyan}{\pmb{x_2}}$}; 

\draw [->, name path global=f3xline] (0,0) -- (\lengthXaxis,0) node[right] {$\pmb{\textcolor{green}{y}}$}; 

\draw[name path global=f3a, thick, color=purple, domain=3:5.5, shift={(-.5,-.4)}] 
       plot (\x,{25*exp(-.7*\x-.25)+.7}) node[left,yshift=-.125cm]{$\bar{u}_2$}; 
\draw[name path global=f3b, thick, color=purple, domain=3:5.5, shift={(-1.5,-.4)}] 
       plot (\x,{25*exp(-.7*\x-.25)+.7});   
\draw[name path global=f3c, thick, color=purple, domain=3:5.5, shift={(.5,-.4)}] 
       plot (\x,{25*exp(-.7*\x-.25)+.7});   
\end{scope} 

\path [name intersections={of=f1a and f1b, by={is f1 left, is f1 right}}];

\path [name path=left vline]% vertical line 
       (is f1 left) -- 
       (is f1 left|-current bounding box.south) 
;

\path [name intersections={of=f2xline and left vline, by=is f2xline left vline}]; 

\draw [dotted] (is f1 left) node[above,xshift=.125cm]{$A$} -- (is f2xline left vline) node[below]{$A’$}; 

\path [name path=right vline]% s.o. 
       (is f1 right) -- 
       (is f1 right|-current bounding box.south) 
; 

\path [name intersections={of=f2xline and right vline, by=is f2xline right vline}]; 

\draw [dotted] (is f1 right) node[above,xshift=.125cm]{$B$}-- (is f2xline right vline) node[below]{$B’$}; 


\draw [name path global=f6] let 
\p1 = ($ (is f2xline left vline) - (is f2xline right vline) $), 
\n2 = {veclen(\x1,\y1)/2} 
in 
(is f2xline right vline)
arc [start angle=0, end angle=180, x radius=\n2, y radius=2*\n2/3] 
coordinate[midway] (j1); 

\path [name intersections={of=f5 and f6, by=i1,total=\t}];
%\pgfextra{\typeout{\t}};
\path [name path=iline]% s.o. 
       (i1|-current bounding box.north) -- 
       (i1|-current bounding box.south) ;

\path [name intersections={of=f1a and iline, by=i2},
name intersections={of=f1b and iline, by=i3},
name intersections={of=f3a and iline, by=i4}];
\draw [dotted] (i2|-0,-12) node[below]{$y^{\ast}$} |- (i2-|0,0) node[left]{$x^{\ast}$}
foreach \X in {1,3,4} {(i\X) node[above]{$C’$} -- (i\X-|0,0)} node[left]{$x_2^{\ast}$};

\path [name path=jline]% s.o. 
       (j1|-current bounding box.north) -- 
       (j1|-current bounding box.south) ;
\path [name intersections={of=f1b and jline, by=j2}];
\draw [dotted] (j2|-0,-6) |- (j2-|0,0) node{$bbb$};  
\end{tikzpicture} 
 


\end{document}