Question

My question (will attach as image) is with respect to PS4-2h4-a. They claim the maximum value of f(x,y) is infinity in the first quadrant and that it goes to infinity along the line y=x. I though that over a bounded region, the maximum/minimum had to occur at either the critical points or along the boundaries. Thanks in advance.

Answer 1

Attached is the question and the answer.

Answer 2

** I thought that over a bounded region, the maximum/minimum had to occur at either the critical points or along the boundaries. **

Yes, but the first quadrant is unbounded (or alternatively, its "boundary" is + infinity in both the x and y directions).

Answer 3

Hey phi. I think I may be confused as to the correct meaning of "boundaries". I thought, and the answer at first seemed to indicate, that the bounds of the first quadrant were the lines y = 0 and x =0 with x>=0 and y>=0. Therefore, the maximum/minimum either had to occur along those lines or at a critical point within the quadrant.

Answer 4

here "boundary" refers to the boundary of the ENTIRE region under consideration...

here, you are considering the region to be the whole positive quadrant...

so the statement the maximum/minimum has to occur at either the critical points or along the boundaries" in this case: means that max/min occur at either the critical points, or at ANY point in the first quadrant.

Answer 5

To amplify on baru 's comment

** I thought that over a bounded region, the maximum/minimum had to occur at either the critical points or along the boundaries. **

This is not a mysterious "rule", but more a summary of common sense. Take the simple case of
y=x from x=0 to x=1

the max/min either occurs at a critical point within the [0,1] interval (and that means the curve changed direction), or at the "edges"

if we make the interval x>=0
then it should be clear there is no absolute max value... as you move to greater values of x, the line continues to rise. We say the max value is +infinity (where infinity is really an "idea" , not a number). If we use interval notation, we could say the interval is
\( [0,+\infty)\) , the "boundary" is at + infinity, and the max value occurs on the "boundary" +infinity