The maximum principle
Continuous functions whose domain is a closed enjoy two useful properties:
- the maximum principle
- the intermediate value theorem
This card covers the first. If a function is continuous and it's domain is a closed interval, then it has a maximum value (one or more values of the domain map to a maximum).
Bounded functions
Let \( X \subseteq \mathbb{R} \) and let \( f : X \to \mathbb{R} \) be a function.
We say that \( f \) is bounded from above iff [...].
We say that \( f \) is bounded from below iff [...].
We say that \( f \) is bounded iff it is both bounded from above and bounded from below.
[...] are bounded
Let \( a < b \) be real numbers. Let \( f : [a,b] \to \mathbb{R} \) be a function [...] on \( [a, b] \). Then \( f \) is a bounded function.
Obtains maximum/minimum
Let \( f: X \to \mathbb{R} \) be a function, and \( x_0 \in X \).
We say that \( f \) obtains its maximum at \( x_0 \) iff \( f(x_0) \ge f(x) \text{ for all } x \in X \).
We say that \( f \) obtains its minimum at \( x_0 \) iff \( f(x_0) \le f(x) \text{ for all } x \in X \).
Maximum principle
Let \( a < b \) be real numbers. Let \( f : [a,b] \to \mathbb{R} \) be a function continuous on \( [a, b] \).
Then there is an \( x_{max} \in [a, b] \) such that \( f(x_{max}) \ge f(x) \text{ for all } x \in [a, b] \) (\( f \) obtains its maximum at \( x_{max} \)).
Similarly, there is an \( x_{min} \in [a, b] \) such that \( f(x_{min}) \le f(x) \text{ for all } x \in [a, b] \) (\( f \) obtains its minimum at \( x_{min} \)).