>

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;

> 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;

> 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;

> 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;

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

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

>