> 
#  MAXIMUM AND MINIMUM VALUES  
# 
# We look for the critical points of several functions
# of two variables.
# 
> with(plots):
# 
# First function. This function has one critical point,
# a local minimum.
> f:=x^2+y^2;

                                   2    2
                             f := x  + y

> plot3d(f,x=-2..2,y=-2..2,color=red);
> contourplot(f,x=-2..2,y=-2..2,color=red);
> 
# Second function: This function has a saddle point.
> g:=x^2-y^2;

                                   2    2
                             g := x  - y

> plot3d(g,x=-2..2,y=-2..2,color=red);
> contourplot(g,x=-2..2,y=-2..2,color=red);
> 
# Third function. This function has local maximum and
# local minima.
# 
> h:=(x^2-4)^2+6*y^2;

                               2     2      2
                        h := (x  - 4)  + 6 y

> plot3d(h,x=-3..3,y=-2..2,color=red);
> contourplot(h,x=-3..3,y=-2..2,color=red);
> 
# Fourth function. We look at this function on a limited domain.
# It achieves its absolute maximum on the boundary of the domain.
# 
> q:=x^2+2*x*y-y;

                               2
                         q := x  + 2 x y - y

> plot3d(q,x=-2..2,y=-2..2,color=red);
> contourplot(q,x=-2..2,y=-2..2,color=red);
>