Question

For the graph of a function y = f(x) shown to the right, find the absolute maximum and the absolute minimum, if they exist. Identify any local maxima or local minima.

Answer 1

For a constant function f(x) = 10 restricted between x = 0 and x = 20, there are no distinct maxima or minima points. The entire line represents both the absolute and local maximum and minimum values of 10.

Step-by-step explanation:

The student wants to find the absolute maximum and minimum, and any local maxima or minima, for a function f(x) within a given range. As described, the function graph is a horizontal line restricted between x = 0 and x = 20, which implies that the function is constant.

Since the function does not change, the entire line represents both the absolute and local maximum and minimum. In this case, with f(x) = 10, there are no distinct maximum or minimum points other than the constant value of the function itself.

There is also no perceived frequency line for a car traveling twice as fast, as this information might be from a different context that is not relevant to the question presented.

Answer 2

The absolute maximum of the function occurs at x = 2, and the absolute minimum at x = 5. There are no local maxima or minima.

Step-by-step explanation:

The graph of the function y = f(x) shows a single absolute maximum at x = 2, where the function reaches its highest value on the graph. This point represents the global peak of the function across its entire domain. Similarly, the absolute minimum occurs at x = 5, where the function reaches its lowest value on the graph, representing the global trough.

However, between these points, there are no local maxima or minima. Local extrema would occur at points where the function reaches a high or low relative to its immediate neighboring points but are not the highest or lowest over the entire domain. In this case, the graph doesn't exhibit any other points where the function locally peaks or troughs between the absolute maximum and minimum. Therefore, there are no local maxima or minima present.

Understanding the concept of absolute and local extrema in a graph helps identify critical points where a function reaches its highest or lowest values. Analyzing these points aids in comprehending the behavior and characteristics of the function over its domain.