For the autocatalytic reaction
-rA=k*cA*(cA0-cA)
 there is a simple analytical expression for the optimum recycle ratio (see Missen et al., "Introduction to Chemical Reaction Engineering and Kinetics," Wiley, 1999, p. 384) and we can use it to verify the graphical solution (see Example 6.3 in Levenspiel):

\(TraditionalForm\`ln\ [\(C\_A0 + \(R\_\(\(opt\)\(\ \)\)\) C\_Af\)\/\(\(R\_opt\) C\_Af\)]\  = \ \(\(C\_A0\)(1 + R\_opt)\)\/\(\(R\_opt\)(C\_A0 + \(R\_opt\) C\_Af)\)\)

cA0 = 2;
func1 = 
        Table[0.05 + 0.02  x, {x, 0, 100}];
func2 = Table[0 + 0.5  x, {x, 0, 40}];
recipRate = 
        Plot[1/(cA*(2 - cA)), {cA, 0.05, 2}, 
          GridLines -> {func1, func2}, 
          PlotStyle -> Thickness[0.003]];