4.2 Exercises on optimization: definitions
- Consider the function f of a single variable defined by f(x) = −x − 1 for x < −1, f(x) = 0 for −1 ≤ x ≤ 1, and f(x) = x − 1
for x > 1. Is the point x = 0 a global maximizer and/or a global minimizer and/or a local maximizer and/or a local minimizer of f?
The point x = 0 is not a global maximizer (f(2) = 1 > f(0) = 0, for example), but is a global minimizer (f(x) ≥ f(0) = 0 for all x), and is both a local maximizer (f(x) ≤ f(0) = 0 for all x with −1 ≤ x ≤ 1) and a local minimizer.
- Consider the function f of a single variable defined by f(x) = x + 1 for x < −1, f(x) = 0 for −1 ≤ x ≤ 1, and f(x) = x − 1 for x
> 1. Is the point x = 0 a global maximizer and/or a global minimizer and/or a local maximizer and/or a local minimizer of f?
The point x = 0 is not a global maximizer (f(2) = 1 > f(0) = 0, for example), or a global minimizer (f(−2) = −1 < f(0) = 0, for example). It is both a local maximizer (f(x) ≤ f(0) = 0 for all x with −1 ≤ x ≤ 1) and a local minimizer.