g[x_,y_]=y-x(x-1)(x-2)

curvas=ContourPlot[g[x,y],{x,0,2},{y,-2,2},ContourShading->False,ContourLabels->True,GridLines->Automatic]

Show[RegionPlot[And[0.5≤x≤1.5,-1≤y≤1-8(((x-1))^2)],{x,0,2},{y,-2,2},GridLines->Automatic],Graphics[{PointSize[Large],Red,Point[{{1.2,-1},{1.2,1-8(((1.2-1))^2)}}],Line[{{1.2,-1},{1.2,1-8(((1.2-1))^2)}}],Text["y = y1[x] = -1",{1.25,-1.15}],Text["y = y2[x] = 1-8(x-1)^2",{1.5,0.85}]}]]

region=RegionPlot[And[0.5≤x≤1.5,-1≤y≤1-8(((x-1))^2)],{x,0,2},{y,-2,2}];

Show[region,curvas]

g[0.5,-1]

g[1,1]

Solve[D[x(x-1)(x-2)+k,x]== D[1-8(((x-1))^2),x],x]

xmax=%[[2,1,2]]

N[xmax]

N[1-8(((xmax-1))^2)]

N[g[xmax,1-8(((xmax-1))^2)]]

minmax=ContourPlot[{g[x,y]==g[0.5,-1],g[x,y]==g[xmax,1-8(((xmax-1))^2)]},{x,0,2},{y,-2,2},ContourStyle->{{Blue,Thick},{Blue,Thick}}];

Show[region,curvas,minmax,Graphics[{PointSize[Large],Red,Point[{{0.5,-1},{xmax,1-8(((xmax-1))^2)}}]}]]