Get["font.math"];
f[x_] := 2x^3 - 9 x^2 + 1;
a =   4.0;
b = 5.0;
g[{a_,b_}] := (
               x=a;y=b;
               If[f[(x+y)/2] >0, y = (x+y)/2, x = (x+y)/2];
               Return[{x,y}]
              );
A = Table[{0,0},{i,1,20}];
A[[1]] = {a,b};
Do[ A[[i+1]] = g[ A[[i]] ],{i,1,19}];
B1 = Table[ {A[[i]][[1]],f[A[[i]][[1]]]},{i,1,19}];
c = ListPlot[B1,PlotStyle->{RGBColor[1,0,0],PointSize[0.003]}];
B2 = Table[ {A[[i]][[2]],f[A[[i]][[2]]]},{i,1,19}];
d = ListPlot[B2,PlotStyle->{RGBColor[1,0,0],PointSize[0.003]}];
e = Plot[f[x],{x,4,5},PlotStyle->{Thickness[0.001]}];
h = Table[ Plot[ 20-i ,{x,A[[i,1]],A[[i,2]]},
     PlotStyle->{RGBColor[1,0,0],Thickness[0.003]}],{i,1,19}];
k = Show[c,d,e,h,
          PlotLabel->Style[Framed["y = 2x^3 - 9 x^2 + 1"],12,
             Blue,Background->Lighter[Yellow]],PlotRange->All];