Question

1. identify any global extrema.
2. identify any local extrema.

use the graph below for 9 - 10.

1. identify any global extrema.
2. identify any local extrema.
3. explain the difference between a global maximum and a local maximum.
4. draw an example of a graph with a global minimum and a local maximum, but no global maximum.
5. draw an example of a graph with local maximums and minimums, but no global extrema.

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:

1. No global extrema.
2. Local maximum at \(y = 10\) (around \(x = 0\)), local minimum (around \(x = 3\)).
3. No global extrema.
4. Local maximum at \(y = 10\) (around \(x = 0\)), local minimum (around \(x = 3\)).
5. A global maximum is the highest value over the entire domain, a local maximum is the highest value in a small neighborhood.
6. 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.
7. Draw a wave - like function like \(y = \sin(x)+x\) that oscillates indefinitely as \(x\to\pm\infty\).