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.


Consider the function defined by f(\(x\)) = –\(x^{2}\) + 4, and represented by the parabola below:

maximum fonction

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.

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