We have the condition

So

Thus we may consider the maximum of the function

where is an arbitrary constant.

Now:

The first equation says

So the second equation now says:

So we have two equations in two unknowns:

Adding the two equations gives

Thus

So there are a total of 4 solution points:

where the signs are independent.

I leave it to you to determine which are relative maxima or relative minima.

-Dan