Notebook 2

f[x_,  y_]  =  x^2*y  +  2  x*y^2  -  3  y  +  y^4

fx[x_,  y_]  =  D[f[x, y],  x]

fy[x_,  y_]  =  D[f[x, y],  y]

critpts  =  
      NSolve[{fx[x,  y]  ==  0,  fy[x, y]  ==  0},  {x,  y}]

fxx[x_,  y_]  =  D[fx[x, y],  x];  
    disc[x_,  y_]  =  
      fxx[x,  y]*D[fy[x,  y],  y]  -  D[fy[x,  y],  x]^2

{fxx[x,  y],  disc[x,  y]}  /.  critpts  //  TableForm

ContourPlot[f[x,  y],  {x,  -3,  3},  {y,  -3,  3},  
      Contours -> 50,  PlotPoints -> 50,  ColorFunction -> Hue];