A point \(x \in [a,b]\) such that \(f(x) \leq f(y)\) for all \(y \in [a,b]\) is where \(f\) achieves its minimum (plural: minima). The value \(f(x)\) is the minimum. This is also called a global or absolute minimum on \([a,b]\text{.}\)
A point \(x \in [a,b]\) such that \(f(x) \geq f(y)\) for all \(y \in [a,b]\) is where \(f\) achieves its maximum (plural: maxima). This is also called a global or absolute maximum on \([a,b]\text{.}\) The value \(f(x)\) is the maximum.
A value \(x\) such that \(f(x) \leq f(y)\) for all \(y\) in some open interval \(I\) containing \(x\text{,}\) which could be a lot smaller than the whole interval \((a,b)\text{,}\) is where \(f\) achieves its local minimum. The terms local maximum and local extremum are defined analogously.
A somewhat subtle but important note about the language here: the input value \(x\) is where the extremum is achieved; the extremum itself is always an output value.
In applications, we sometimes want to know just what the maximum is (an output); sometimes just where the maximum occurs (an input); and sometimes both. By way of example:
If I want to build a building to house my flying squirrels, I need to know what the maximum height they’re capable of flying is, but I don’t really care when they get to that height.
If I need to build a window which admits the most possible light, what I care about is how to set the dimensions (an input), but the amount of light actually let in (in lumens, say) isn’t really needed.
If I’m running a widget factory and I want to know what production level will maximize my profit, the input where the maximum occurs (a number of widgets per hour) is important, but for fiscal planning I also need to know what that maximum (a number of dollars) actually is.
As you may have noticed, we’ll reserve the word what to refer to the extremum itself (the output value), and talk about when, where, or how that extremum occurs.
Let \(f\) be a continuous function on the closed interval \([a,b]\text{.}\) Then \(f\) has at least one absolute minimum on \([a,b]\) and at least one absolute maximum on \([a,b]\text{.}\)
