Absolute Maximum and Minimum Values of a Function
Big Idea: Finding the absolute (global) maximum and minimum values of a function means identifying the highest and lowest points the function reaches over a given interval. These are essential in optimization — one of the most powerful applications of calculus.
1. Formal Definitions
Absolute Maximum: A function $f$ has an absolute maximum at $x = c$ on an interval $[a, b]$ if:
$$f(c) \geq f(x) \quad \text{for all } x \in [a, b]$$
The value $f(c)$ is called the absolute maximum value.
Absolute Minimum: A function $f$ has an absolute minimum at $x = c$ on an interval $[a, b]$ if:
$$f(c) \leq f(x) \quad \text{for all } x \in [a, b]$$
The value $f(c)$ is called the absolute minimum value.