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Minimums and Maximums

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Maximums and Minimums

When riding a roller coaster there is always one point that is the absolute highest off the ground. There are usually many other places that reach fairly high, just not as high as the first. How do you identify and distinguish between these different peaks in a precise way? 

Guidance

A global maximum refers to the point with the largest  y value possible on a function. A global minimum refers to the point with the smallest  y value possible. Together these two values are referred to as global extrema. There can only be one global maximum and only one global minimum. Global refers to entire space where the function is defined. Global extrema are also called absolute extrema.

In addition to global maximums and global minimums, there are also local extrema or relative maximums and relative minimums. The word relative is used because in relation to some neighborhood, these values stand out as being the highest or the lowest. 

Calculus uses advanced analytic tools to compute extreme values, but for the purposes of PreCalculus it is sufficient to be able to identify and categorize extreme values graphically or through the use of technology. For example, the TI-84 has a maximum finder when you select <2 nd > then <trace>.

Example A

Identify and categorize all extrema:

Solution: Since the function appears from the arrows to increase and decrease beyond the display, there are no global extrema. There is a local maximum at approximately (0, 3) and a local minimum at approximately (2.8, -7). 

Example B

Identify and categorize all extrema:

Solution: Since the function seems to abruptly end at the end points and does not go beyond the display, the endpoints are important.

There is a global minimum at (0, 0). There is a local maximum at (-1, 1) and a global maximum at (5, 5). 

Example C

Identify and categorize all extrema.

Solution: Since this function appears to increase to the right as indicated by the arrow there is no global maximum. There are not any other high points either, so there are no local maximums. There is only the end point at (0, 0) which is a global minimum.

Concept Problem Revisited

Maximums and minimums should be intuitive because they simply identify the highest points and the lowest points, or the peaks and the valleys, in a graph. There is a formal distinction about whether a maximum is the highest on some local open interval (does not matter how small), or whether it is simply the highest overall. 

Vocabulary

Global extrema and absolute extrema are synonyms that refer to the points with the  y values that are either the highest or the lowest of the entire function.

Local extrema and relative extrema are synonyms that refer to the points with the  y values that are the highest or lowest of a local neighborhood of the function.

Guided Practice

1. Identify and categorize all extrema. 

2. Identify and categorize the extrema. 

3. Identify and categorize the extrema. 

Answers:

1. There are no global or local maximums or minimums. The function flattens, but does not actually reach a peak or a valley. 

2. There are no global extrema. There appears to be a local maximum at (0, 0) and a local minimum at (1, -1). 

3. There are no global extrema. There appears to be a local maximum at (-1.2, 5.3) and a local minimum at (1.2, -1.8). These values are approximated. If a function was given, you would need to graph the function on a calculator and use the maximum and minimum features to identify more exact points.

Practice

Use the graph below for 1-2.

1. Identify any global extrema.

2. Identify any local extrema.

Use the graph below for 3-4.

3. Identify any global extrema.

4. Identify any local extrema.

Use the graph below for 5-6.

5. Identify any global extrema.

6. Identify any local extrema.

Use the graph below for 7-8.

7. Identify any global extrema.

8. Identify any local extrema.

Use the graph below for 9-10.

9. Identify any global extrema.

10. Identify any local extrema.

11. Explain the difference between a global maximum and a local maximum.

12. Draw an example of a graph with a global minimum and a local maximum, but no global maximum.

13. Draw an example of a graph with local maximums and minimums, but no global extrema.

14. Use your graphing calculator to identify and categorize the extrema of:

       f(x)=\frac{1}{2} x^4+2x^3-6.5x^2-20x+24.

15. Use your graphing calculator to identify and categorize the extrema of:

      g(x)=-x^4+2x^3+4x^2-2x-3.

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