Question

find the absolute extreme values and where they occur.
select the correct choice below and, if necessary, fill in the answer boxes within your choice.
a. the absolute maximum value is at x = there is no absolute minimum value.
b. the absolute minimum value is at x = there is no absolute maximum value.
c. the absolute maximum value is at x =, and the absolute minimum value is at x =
d. there is no absolute maximum value or absolute minimum value.

Explanation:

Step1: Identify the maximum and minimum values from the graph

The graph has a closed - dot (which represents a point that is included in the function's domain and range) and an open - dot (which represents a point that is not included). The \(y\) - value of the closed - dot is higher than the \(y\) - values of the other points on the graph. The open - dot means that the lower \(y\) - value is not actually attained by the function.

The closed - dot has coordinates \((- 4,-2)\) and the open - dot has coordinates \((2,-6)\). Since the open - dot is not part of the function (the function does not actually reach that \(y\) - value), we consider the \(y\) - value of the closed - dot as the maximum (because there are no other points with a higher \(y\) - value that are included in the function).

Step2: Analyze the options

Option A: The absolute maximum value is \(-2\) at \(x=-4\). There is no absolute minimum value (because the lower \(y\) - value corresponding to the open - dot is not attained by the function).

Answer:

A. The absolute maximum value is \(-2\) at \(x =-4\). There is no absolute minimum value.