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Maximize $f(x,y) = x + \sqrt{y}$ subject to $x+y \leq 1$ such that $x, y \geq 0$.

I read that if the region is closed and bounded, then there will be more than one critical point. However, solving it gives only one point. Why is it so?

dictatemetokcus
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  • Well a critical point is a point such that $\nabla f =0$, this happens only one you have an interior critical point. It may well be that your function has no critical points in the interior (think of an affine function) but clearly attains a maximum and a minimum at the boundary – tommy1996q Feb 24 '22 at 12:53

1 Answers1

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For that thing you read to be correct, you need to include boundary points among the points you check. I would not call them "critical points", but perhaps some textbooks do.

So: You find the critical points of $f$ in the interior of the triangle. Then you check also the points on the boundary of the triangle. Unless it is obvious, perhaps you do a maximum/minimum computation on each of the three edges. And then you do the three vertices. You end up with a list of candidate points; The function $f$ will achieve its maximum in at least one of them, and will achieve its minimum in at least one of them.

Since your function is not constant on the triangle, the maximum and minimum will be achieved at different points. That is why there must be at least two of them.

GEdgar
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