## Absolute Maximum

### The highest point over the entire domain of a function or relation.

The absolute maximum of a function is the highest value that the function attains over its entire domain. It represents the largest possible output value of the function, and there can only be one absolute maximum.

To find the absolute maximum of a function, we need to examine every point on the function’s domain and determine which point has the highest output value. This can be done by using calculus techniques such as differentiation and optimization or by examining the function visually using a graph.

For example, the function f(x) = x^2 – 4x + 5 has a domain of all real numbers. We can find the absolute maximum of this function by taking its derivative: f'(x) = 2x – 4. Setting f'(x) = 0, we get x = 2. We can then evaluate the function at this point to find its output value: f(2) = 1. Therefore, the absolute maximum of f(x) is 1, and it occurs at x = 2.

Note that a function may not have an absolute maximum if it is unbounded or if its domain is restricted. Additionally, a function can have multiple local maxima (points where the function reaches a local high point), but only one absolute maximum.

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