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Study Guides > College Algebra

Use a graph to locate the absolute maximum and absolute minimum

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 y-y\text{-} 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 f(x)=x3f\left(x\right)={x}^{3} is one such function.

A General Note: Absolute Maxima and Minima

The absolute maximum of ff at x=cx=c is f(c)f\left(c\right) where f(c)f(x)f\left(c\right)\ge f\left(x\right) for all xx in the domain of ff.

The absolute minimum of ff at x=dx=d is f(d)f\left(d\right) where f(d)f(x)f\left(d\right)\le f\left(x\right) for all xx in the domain of ff.

Example 10: Finding Absolute Maxima and Minima from a Graph

For the function ff shown in Figure 11, find all absolute maxima and minima.
Graph of a polynomial.Figure 11

Solution

Observe the graph of ff. The graph attains an absolute maximum in two locations, x=2x=-2 and x=2x=2, because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate at x=2x=-2 and x=2x=2, which is 1616.

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

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