4.3 Existence of an optimum
Before we start to think about how to find the solution to a problem, we need to think about whether the problem has a solution. Here are some specifications of f and S for which the problem does not have any solution.
- f(x) = x, S = [0, ∞) (i.e. S is the set of all nonnegative real numbers). In this case, f increases without bound, and never attains a maximum.
- f(x) = 1 − 1/x, S = [1, ∞). In this case, f converges to the value 1, but never attains this value.
- f(x) = x, S = (0, 1). In this case, the points 0 and 1 are excluded from S (which is an open interval). As x approaches 1, the value of the function approaches 1, but this value is never attained for values of x in S, because S excludes x = 1.
- f(x) = x if x < 1/2 and f(x) = x − 1 if x ≥ 1/2; S = [0, 1]. In this case, as x approaches 1/2 the value of the function approaches 1/2, but this value is never attained, because at x = 1/2 the function jumps down to −1/2.
For functions of many variables, we need to define the concept of a bounded set.
- Definition
- The set S is bounded if there exists a number k such that the distance of every point in S from the origin is at most k.
- Example
- The set [−1, 100] is bounded, because the distance of any point in the set from 0 is at most 100. The set [0, ∞) is not bounded, because for any number k, the number 2k is in the set, and the distance of 2k to 0 is 2k which exceeds k.
- Example
- The set {(x, y): x^{2} + y^{2} ≤ 4} is bounded, because the distance of any point in the set from (0, 0) is at most 2.
- Example
- The set {(x, y): xy ≤ 1} is not bounded, because for any number k the point (2k, 0) is in the set, and the distance of this point from (0, 0) is 2k, which exceeds k.
- Proposition (Extreme value theorem)
- Let f be a function of many variables defined on a set X and let S be a subset of X. If f is continuous and S is compact then the problems of maximizing and minimizing f(x) subject to x ∈ S both have solutions.
Note also that the result gives only a sufficient condition for a function to have a maximum. If a function is continuous and is defined on a compact set then it definitely has a maximum and a minimum. The result does not rule out the possibility that a function has a maximum and/or minimum if it is not continuous or is not defined on a compact set. (Refer to the page on logic if you are unclear on this point.)