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