11.7 Maximum and Minimum Values

Critical Points for Multi-Variable Functions

In the same way how a critical point for a single-variable function occurs when f’(x) = 0, so too do critical points occur for multivariable functions when the gradient vector (i.e. ALL of the partial derivatives) equal to 0.

However, simply because a critical point exists does not guarantee that a maximum or minimum will exist. However, in order for an extrema TO exist, it must also be a critical point.

Second Derivative Test for Multi-Variable Functions

Absolute Extrema

In the same way how absolute extrema required that we test both the critical points within [a,b] AND [a,b] itself, we must test the critical points inside of a disk D AND the endpoints of that bounded region (assuming inclusivity).

Closed Set

A closed set is a disk which includes its endpoints; in terms of inequalities, it would be written as >=. That sort of thing.

Bounded Set

A Bounded set is a set whereby all of its values are contained within a disk.