# Maximum of a Function

## 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($$x$$) 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($$x$$) = –$$x^{2}$$ + 4, and represented by the parabola below:

If $$x = 0$$, then $$f(x) = 4$$.
For any other value of $$x$$, $$f(x) < 4$$.
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 $$f(x) = x^{2} + 4$$, does not have a maximum, but its minimum is 4.