7. identify any global extrema.\n8. identify any local extrema.\nuse the graph below for 9 - 10.\n9…

Explanation:

Step1: Define global extrema

Global maximum is the highest value of the function over its entire domain, global minimum is the lowest.

Step2: Analyze the given graph

From the graph, as (x) approaches (\pm\infty), (y) approaches (\infty), so there is no global maximum. The lowest - point on the graph in the visible domain is at the local minimum point in the interval shown. But since the function goes downwards as (x\to-\infty), there is no global minimum. So, no global extrema.

Step3: Define local extrema

Local maximum is a point where the function value is greater than the values at nearby points, and local minimum is a point where the function value is less than the values at nearby points.

Step4: Identify local extrema on the graph

The function has a local maximum at the point where (y = 10) (around (x = 0)) and a local minimum at the point where the function bottoms - out in the positive (x) region (around (x = 3)).

Step5: Explain the difference between global and local maximum

A global maximum is the highest value of the function over its entire domain. A local maximum is the highest value of the function in a small neighborhood around a point. A local maximum may not be the highest value overall.

Step6: Draw a graph with global minimum and local maximum but no global maximum

Draw a parabola opening downwards with a vertex (local maximum) and then a part of the graph that goes downwards indefinitely as (x\to\pm\infty). The lowest - point of the part of the graph considered is the global minimum.

Step7: Draw a graph with local maxima and minima but no global extrema

Draw a wave - like function that oscillates indefinitely in both directions as (x\to\pm\infty). For example, (y=\sin(x)+x). It has local maxima and minima but as (x\to\pm\infty), (y\to\pm\infty) so there are no global extrema.

Answer:

7. No global extrema.
8. Local maximum at (y = 10) (around (x = 0)), local minimum (around (x = 3)).
9. No global extrema.
10. Local maximum at (y = 10) (around (x = 0)), local minimum (around (x = 3)).
11. A global maximum is the highest value over the entire domain, a local maximum is the highest value in a small neighborhood.
12. Draw a parabola opening downwards with a vertex (local maximum) and a part of the graph going downwards indefinitely as (x\to\pm\infty) to get a global minimum.
13. Draw a wave - like function like (y = \sin(x)+x) that oscillates indefinitely as (x\to\pm\infty).