Find the absolute maximum and minimum values of f on the set d. f(x, y) = x3 − 3x − y3 + 12y + 1, d is quadrilateral whose vertices are (−2, 3), (2, 3), (2, 2), and (−2, −2). absolute maximum value minimum value

is the set of points,

Check for critical points:

Of these 4 points, only 2 belong to , (-1, 2) and (1, 2), for which we have

Now look for extrema along the boundary.

We have for and for , which indicates a local maximum at and minima at the endpoints of this boundary. So

We have for , so we have extrema at the endpoints of this boundary.

which tells us is strictly increasing on this boundary, giving the extrema we already know about,

We have for and , and for . This indicates a maximum at and a minimum at , with

From this analysis, we find that attains an absolute maximum of 19 at (-1, 2) and (2, 2), and an absolute minimum of -17 at (-2, -2).

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