# Determine Where a Function is Increasing, Decreasing, or Constant

## 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 - 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 13.

**Figure 13**

Not every function has an absolute maximum or minimum value. The toolkit function is one such function.

#### ABSOLUTE MAXIMA AND MINIMA

The **absolute maximum** of at is where for all in the domain of . The **absolute minimum** of at is where for all in the domain of .

#### EXAMPLE 10

##### Finding Absolute Maxima and Minima from a Graph

For the function shown in Figure 14, find all absolute maxima and minima.

**Figure 14**

##### Solution

Observe the graph of . The graph attains an absolute maximum in two locations, and , because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the -coordinate at and , which is 16.

The graph attains an absolute minimum at , because it is the lowest point on the domain of the function's graph. The absolute minimum is the -coordinate at , which is .