α[t_]={1+0.5 Cos[t],1+0.3 Sin[t]};

lips=ParametricPlot[α[t],{t,0,2π},PlotStyle->{Green,Thick}];

u[x_,y_]=3x+5y;

iso=ContourPlot[u[x,y],{x,0,2},{y,0,2},ContourShading->False,ContourLabels->True,GridLines->Automatic,Contours->21];

Show[iso,lips]

θ[t_]=u[α[t][[1]],α[t][[2]]]

D[%,t]

θ[π/4]

θ[5π/4]

minmax=ContourPlot[{u[x,y]==u[α[π/4][[1]],α[π/4][[2]]],u[x,y]==u[α[5π/4][[1]],α[5π/4][[2]]]},{x,0,2},{y,0,2},ContourStyle->{{Thick,Blue},{Thick,Blue}}];

Show[iso,lips,minmax,Graphics[{PointSize[Large],Red,Point[{α[π/4],α[5π/4]}]}]]

Grad[u[x,y],{x,y}]

Show[iso,lips,Graphics[{PointSize[Large],Red,Point[{α[π/4],α[5π/4]}],Arrow[{{α[π/4],α[π/4]+(0.6)α'[π/4]},{α[5π/4],α[5π/4]+(0.6)α'[5π/4]},{α[π/4],α[π/4]+(0.1){3,5}},{α[5π/4],α[5π/4]+(0.1){3,5}}}]}]]