Find absolute maximum and minimum

Question

Find the absolute maximum and absolute minimum of the function $f(x,y)=xy-5y-25x+125$ on the region above $y=x^2$ and on or below $y=27$.

My solution:

Critical point: $(5, 25)$

$f$ has a saddle point at $(5, 25)$ so it can't be used to find absolute min/max (correct me if i'm wrong).

Boundary points: $(-5, 0), (5, 0), (0,27)$

$f$ has absolute max of $250$ at $(-5, 0)$

$f$ has absolute min of $-10$ at $(0, 27)$

These are my answers, but the online system (homework portal) doesn't accept them so they may be wrong. Any ideas?

Answer 1

You have the wrong boundary points. The curves $y=x^2$ and $y=27$ intersect at $(\pm 3\sqrt{3},27)$. The points $(\pm 5,0)$ are not in the region at all.

And since you are finding the extrema of a function of two variables, you need to consider points on the entire boundary of the region, not just the "boundary points." So you also need to look at $(x,27)$ for $-3\sqrt{3} <x <3\sqrt{3}$. You many also need to look at $(x,x^2)$ for the same $x$'s, but since the region is above $y=x^2$ probably not.