Cubic equation and tangent circle

First define a parametric curve and regions above and below the curve:

ClearAll[curve, disk, region1,  region2, radius]

curve[x_] := {x, (x + 1) (x - 1) (x - 2)}

region1 = ImplicitRegion[(x + 1) (x - 1) (x - 2) >= y, {{x, -20, 30}, {y, -80, 80}}];

region2 = ImplicitRegion[(x + 1) (x - 1) (x - 2) <= y, {{x, -20, 30}, {y, -80, 80}}];

and a disk with radius r tangent to the curve at curve[t]:

disk[dir_ : -1][t_, r_] := Module[{rr = Rationalize[r, 0],tt = Rationalize[t, 0]}, 
  Disk[curve[tt] + dir rr Cross@Normalize[curve'[tt]], rr]]

For given input region reg and parameter value t find the maximal radius r such that the disk with radius r tangent to the curve at curve[t] stays within reg:

radius[reg_, dir_ : -1][t_?NumericQ] :=
    NMaxValue[{r, RegionWithin[reg, disk[dir][t, r]]}, r]

Examples:

radius[region1][-1]

0.992403  

radius[region1][3/2]

 2.29173 

radius[region2, 1][1]

 0.555255  

pp = ParametricPlot[curve[x], {x, -2, 3}, ImageSize -> Medium, 
   PlotRange -> {{-5, 5}, {-6, 5}}];

frames = Table[Show[pp, 
    Graphics @ {PointSize[Large], Red, Point[curve[t]], FaceForm[], 
      EdgeForm[Red], disk[][t, radius[region1][t]]}], 
  {t, -3/2, 3/2, 1/100}];

Export["diskoncurve.gif", frames]

ParametricPlot[curve[x], {x, -2, 3}, AspectRatio -> Automatic, 
  Epilog -> {FaceForm[], 
    EdgeForm[Red], disk[][1 + 5/10, radius[region1][1 + 5/10]], 
    EdgeForm[Green], disk[][1, radius[region1][1]], 
    EdgeForm[Orange], disk[][1/5, radius[region1][1/5]], 
    EdgeForm[{Dashed, Red}], disk[1][1 + 5/10, radius[region2, 1][1 + 5/10]], 
    EdgeForm[{Dashed, Green}], disk[1][1, radius[region2, 1][1]], 
    EdgeForm[{Dashed, Orange}], disk[1][1/5, radius[region2, 1][1/5]]}]

Edit

f[x_] = (x + 1) (x - 1) (x - 2);
r[x_] = {x, f[x]};
eq1 = r[x1] + t1*Cross@Normalize[r'[x1]];
eq2 = r[x2] + t2*Cross@Normalize[r'[x2]];
plot = ParametricPlot[r[x], {x, -2, 3}, PlotStyle -> Blue];
xmin = x /. Solve[f'[x] == 0, x] // Last;
ymin = f[xmin];
xmax = x /. Solve[f'[x] == 0, x] // First;
ymax = f[xmax];
circles1 = 
  Graphics[Table[{Orange, Circle[eq1, Abs@t1]} /. 
     NMinimize[{0, t1^2 == t2^2, -2 < x1 < xmin < x2 < 3, eq1 == eq2, 
        eq1[[2]] == eq2[[2]] == d}, {x1, x2, t1, t2}][[2]], {d, 
     ymin + 1.5, ymin + 5, .2}]];
circles2 = 
  Graphics[Table[{Green, Circle[eq1, Abs@t1]} /. 
     NMinimize[{0, t1^2 == t2^2, -2 < x1 < xmax < x2 < 3, eq1 == eq2, 
        eq1[[2]] == eq2[[2]] == d}, {x1, x2, t1, t2}][[2]], {d, 
     ymax - 1, ymax - 5, -.2}]];
Show[plot, circles1, circles2, AspectRatio -> Automatic]

Original

Using the method from Find the equidistance curve between two curves

f[x_] = (x + 1) (x - 1) (x - 2);
r[x_] = {x, f[x]};
eq1 = r[x1] + t1*Cross@Normalize[r'[x1]];
eq2 = r[x2] + t2*Cross@Normalize[r'[x2]];
plot = ParametricPlot[r[x], {x, -2, 3}];
Manipulate[
 sol = NMinimize[{0, eq1 == eq2, t1^2 == t2^2, x2 - x1 == c}, {x1, x2,
     t1, t2}, Method -> Automatic];
 disk = Graphics[{Cyan, Opacity[0.2], EdgeForm[Red], 
    Disk[eq1, Abs@t1] /. Last[sol]}];
 Show[plot, disk, AspectRatio -> Automatic], {c, -2.5, 2.5, .01}, 
 ControlPlacement -> Bottom]