how to find absolute max and min

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\text{-}[/latex] coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively.

To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.

Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).

Figure 10

Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\left(x\right)={x}^{3}[/latex] is one such function.

A General Note: Absolute Maxima and Minima

The absolute maximum of [latex]f[/latex] at [latex]x=c[/latex] is [latex]f\left(c\right)[/latex] where [latex]f\left(c\right)\ge f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].

The absolute minimum of [latex]f[/latex] at [latex]x=d[/latex] is [latex]f\left(d\right)[/latex] where [latex]f\left(d\right)\le f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].

Example 10: Finding Absolute Maxima and Minima from a Graph

For the function [latex]f[/latex] shown in Figure 11, find all absolute maxima and minima.

Graph of a polynomial.

Figure 11

Solution

Observe the graph of [latex]f[/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[/latex] and [latex]x=2[/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate at [latex]x=-2[/latex] and [latex]x=2[/latex], which is [latex]16[/latex].

The graph attains an absolute minimum at [latex]x=3[/latex], because it is the lowest point on the domain of the function's graph. The absolute minimum is the y-coordinate at [latex]x=3[/latex], which is [latex]-10[/latex].