(820,#33) It is impossible for a continuous function of one variable to have two local maxima and no local minimum. But there do exist functions of two variables with this property. Show that the function has only two critical points, but has local maxima at both of them. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.

Solution: Using Maple we find two critical points, one at (1,2) the other at (-1,0):

f:=(x,y)->-(x^2-1)^2-(x^2*y-x-1)^2:

solve({diff(f(x,y),x)=0,diff(f(x,y),y)=0},{x,y});

{y = 2, x = 1}, {y = 0, x = -1}

Using the Second Derivative Test,

Both (-1,0) and (1,2) yield local maxima.

The fact that these two points are local maxima is not easy to see from a graph:

No Maple Code for this problem.

by:sh