Question

for this question, please note that end - points are considered relative extrema. please use a comma to separate all answers.
the ordered pair(s) that represent the relative min is/are:
the ordered pair(s) that represent the absolute min is/are:
the ordered pair(s) that represent the relative max is/are:
the ordered pair(s) that represent the absolute max is/are:
if there is more than one answer, enter them separated by a comma.
if there is no answer, enter \dne\
question help: message instructor

Explanation:

Relative Min

A relative minimum is a point where the function changes from decreasing to increasing. Looking at the graph, the point \((1,1)\) is a relative minimum. Also, endpoints can be relative extrema. But here, \((0,2)\) is higher than \((1,1)\) and \((5,0)\) is lower but is not a relative minimum (since the function is decreasing before \(x = 5\)).

Absolute Min

The absolute minimum is the lowest - valued point in the entire domain of the function. Comparing the \(y\) - values of the points \((0,2)\), \((1,1)\), \((4,4)\), \((5,0)\), the \(y\) - value of \(1\) at \((1,1)\) and \(0\) at \((5,0)\). Since \(0<1\), the absolute minimum is \((5,0)\).

Relative Max

A relative maximum is a point where the function changes from increasing to decreasing. The point \((4,4)\) is a relative maximum. Also, endpoints: \((0,2)\) is a local maximum (as the function is decreasing just after \(x = 0\))

Absolute Max

The absolute maximum is the highest - valued point in the entire domain of the function. Comparing the \(y\) - values of the points \((0,2)\), \((1,1)\), \((4,4)\), \((5,0)\), the \(y\) - value of \(4\) at \((4,4)\) is the highest.

Answer:

The ordered pair(s) that represent the relative min is/are: \((1,1)\)
The ordered pair(s) that represent the absolute min is/are: \((5,0)\)
The ordered pair(s) that represent the relative max is/are: \((0,2),(4,4)\)
The ordered pair(s) that represent the absolute max is/are: \((4,4)\)