UC SCI 111
Demo (local) extrema

<< Graphics`ImplicitPlot`

F = x^3 - 3*x^2*y/2 + 6*x^2 + y^2/2
Fx = D[F, x]
Fy = D[F, y]

Solve[{Fx == 0, Fy == 0}, {x, y}]

ContourPlot[F, {x, -2, 3}, {y, -1, 7}, 
      Contours -> 20]

F = x*y*(8*x - y^2 - x^2)
Fx = D[F, x]
Fy = D[F, y]

Solve[{Fx == 0, Fy == 0}, {x, y}]

F = x^2 - y^2
G = x^2 + (y + 2)^2

p1 = 
        ContourPlot[F, {x, -2, 2}, {y, -4, 1}, Contours -> 20, 
          DisplayFunction -> Identity];
p2 = 
        ImplicitPlot[G == 4, {x, -2, 2}, {y, -4, 1}, 
          PlotStyle -> Hue[0.3], 
          DisplayFunction -> Identity];
Show[p1, p2, DisplayFunction -> $DisplayFunction]