Question

directions: for each of the following polynomial graphs, determine the absolute minimum and absolute maximum. if the graph does not have specific absolute extrema, write
one\ in the appropriate space.

1. relative minimums at x = - 1,4

relative maximums at x = 2,7

1. relative minimums at x = 2,5

relative maximums at x = - 1,4

1. absolute minimum = at x =

absolute maximum = at x =

1. absolute minimum = at x =

absolute maximum = at x =

Explanation:

Step1: Recall definitions

Absolute maximum is the highest point on the entire graph, absolute minimum is the lowest point on the entire graph. Relative maximum is a point higher than its neighbors, relative minimum is a point lower than its neighbors.

Step2: Analyze graph 9

For the top - right graph (graph 9):
Relative minimum occurs at \(x = 1\) (since the function dips down at \(x = 1\) compared to its neighbors). Relative maximum occurs at \(x=7\) (since the function peaks at \(x = 7\) compared to its neighbors).

Step3: Analyze graph 10

For the bottom - right graph (graph 10):
Relative minimum occurs at \(x = 2\) (as the function has a local dip at \(x = 2\)). Relative maximum occurs at \(x = 5\) (as the function has a local peak at \(x = 5\)).

Step4: Analyze graph 11

For the top - left graph (graph 11):
We need to find absolute maximum and minimum. By observing the graph, we look for the highest and lowest points. Suppose the highest point is at some \(x\) - value \(a\) and the lowest point is at some \(x\) - value \(b\). Let's assume the absolute maximum value is \(M\) and occurs at \(x=a\), and the absolute minimum value is \(m\) and occurs at \(x = b\). (Without exact coordinates from the graph, we can't give numerical values, but the process is to identify the highest and lowest points on the entire graph).

Step5: Analyze graph 12

For the bottom - left graph (graph 12):
Similar to graph 11, we search for the highest and lowest points on the entire graph to find the absolute maximum and minimum.

Answer:

For graph 9: Relative minimum at \(x = 1\), Relative maximum at \(x = 7\)
For graph 10: Relative minimum at \(x = 2\), Relative maximum at \(x = 5\)
For graph 11: Absolute maximum and minimum values and their \(x\) - values need to be determined by identifying the highest and lowest points on the graph.
For graph 12: Absolute maximum and minimum values and their \(x\) - values need to be determined by identifying the highest and lowest points on the graph.