# Minimum of a Function

## Minimum of a Function

A function $$f$$ defined on a subset E of real numbers has a minimum m at a point $$a$$ in E if m = $$f(a)$$ and if, for all $$x$$ in E, $$f(x)$$ is greater than or equal to $$f(a)$$.

Therefore, m is the minimum 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 minimum of the image of this function is 4.
We can also say that 4 is the minimum of the function f.

The function defined by $$f(x)$$ = –$$x^{2}$$+ 4, does not have a minimum, but its