a=Show[ContourPlot[2 (x^3)+2(y^3)-9 x y==0,{x,-3,3},{y,-3,3},GridLines->Automatic],Graphics[{PointSize[Large],Blue,Thick,Point[{1,2}]}]]

b=ContourPlot[2 (x^3)+2(y^3)-9 x y==0,{x,0.5,1.5},{y,1.5,2.5},GridLines->Automatic,ContourStyle->{Blue,Thick}];

Show[a,b]

c=Plot[2+(4/5)(x-1),{x,0,2},PlotStyle->{Thick,Blue}];

Show[a,c]

D[2 (x^3)+2((y[x])^3)-9 x y[x],x]

Solve[%==0,y'[x]]

%[[1,1,2]]

%/.{x->1,y[x]->2}

f[x_,y_]=2 (x^3)+2(y^3)-9 x y;

gradf[x_,y_]=Grad[f[x,y],{x,y}]

%/.{x->1,y->2}

%.{x,y}

%/.{x->1,y->2}

d=ContourPlot[-4x+5y==6,{x,0,2},{y,6/5,14/5},ContourStyle->{Blue,Thick}];

Show[a,d,Graphics[{PointSize[Large],Thick,Black,Arrow[{{1,2},{1,2}+(1/15)gradf[1,2]}],Text[Style["∇F(1,2)",Large],{-0.6,2.6}],Text[Style["(1,2)",Large],{1.3,1.6}],Text[Style["-4x + 5y = 6",Bold],{-0.5,1.3}],Text[Style["2(x^3)+ 2(y^3) - 9 x y = 0",Bold],{1.3,-1}]}]]