Multivariate Calculus - Bruce E. Shapiro

Lecture 14
Unconstrained Optimization
Every continuous function of two variables f(x, y) that is defined on a closed,
bounded set attains both a minimum and a maximum. The process of finding these
points is called unconstrained optimization. The process of finding a maximum
or a minimum subject to a constraint will be discussed in the following section.
Definition 14.1 Let f(x, y) be a function defined on some set S, and let (a, b) ∈ S
be a point in S. Then we say that
1. f(a, b) is a global maximum of f on S if f(a, b) ≥ f(x, y) for all (x, y) ∈ S.
2. f(a, b) is a global minimum of f on S if f(a, b) ≤ f(x, y) for all (x, y) ∈ S.
3. f(a, b) is a global extremum if it is either a global maximum or a global
minimum on S
Theorem 14.1 Let f(x, y) be a continuous function on some closed, bounded set
S ⊂ R 2 . Then f(x, y) has both a global maximum value and a global minimum value
on S.
The procedure for finding the extrema (maxima and minima) of functions of two
variables is similar to the procedure for functions of a single variable:
1. Find the critical points of the function to determine candidate locations for the
extrema (in the single-dimensional case, these were points where the derivative
is zero or undefined, and the boundary points of the interval);
2. Examine the second derivative at the candidate points that do not lie on the
border (in one-dimension, we had f ′′ (a) < 0 at local maxima and f ′′ (a) > 0
at local minima).
3. Compare internal extrema with the value of the function on the boundary
points and at any points where the derivative or second derivative is undefined
to determine absolute extrema.
Definition 14.2 Let f(x, y) be defined on some set J ⊂ R 2 . Then the candidate
extrema points occur at
101