Maximum of a Function
A function f defined on a subset E of real numbers has a maximum M at a point a in E if M = f(a) and if, for all x in E, f([latex]x[/latex]) is less than or equal to f(a).
Therefore, M is the maximum of the set of images of f.
Example
Consider the function defined by f([latex]x[/latex]) = –[latex]x^{2}[/latex] + 4, and represented by the parabola below:
If [latex]x = 0[/latex], then [latex]f(x) = 4[/latex].
For any other value of [latex]x[/latex], [latex]f(x) < 4[/latex].
Therefore, the maximum of the image of the function is 4.
We can also say that 4 is the maximum of the function f.
The function defined by [latex]f(x) = x^{2} + 4[/latex], does not have a maximum, but its minimum is 4.