1. Chapter Three: Section One Extrema on an Interval

2. Chapter Three: Section One Extrema is a peculiar word, isn’t it? What do we mean when we talk about extrema on an interval. This word is the plural of the similarly odd word extremum. Basically, what we are talking about are minimum or maximum values of a function on a domain interval.

3. Chapter Three: Section One If a function is continuous on a closed domain interval [ a, b ], then that function has both a minimum and a maximum on that interval. In other words, whenever you are looking at a contained, closed piece of a function there has to be a greatest y -value and a least y -value. These values can be repeated, there are cases, such as cosine and sine curves, where the minimum and maximum values are routinely repeated.

4. Chapter Three: Section One So, since we are in the midst of studying derivatives, why would we be talking about minimum and maximum values of a function? How are derivatives related? Using the image of the cosine or sine curves, what can we say about the derivative of a function at the point where the curve attains an extreme value? Think before moving on…

5. Chapter Three: Section One Look at the following picture and think about where the maximum and minimum intervals occur on this region:

6. Chapter Three: Section One This is a continuous function and it is restricted to a closed interval [ a,b ] Where are the extreme values for this function? I chose this example because there are two decoys in the picture.

7. Chapter Three: Section One In this case, the maximum value is obtained at the far-right endpoint marked at x = b and the minimum value is obtained at x = a. There are two points in this interval where a change in direction occurs and these are the points where you might naturally be inclined to look for maximum or minimum values. In fact, we can state the following now as our guideline;  Extrema on a closed interval of a continuous function can occur at the endpoints of the interval or at critical values of the interval

8. Chapter Three: Section One So now we have a new word to define. What is a critical value for a function? A critical value is defined as follows;  If a function f is defined at a domain value c, then c is a critical value of f if f ‘ (c) = 0 or if f ‘ (c) is undefined.

9. Chapter Three: Section One Lets look at that earlier picture again. Try to convince yourself why all 4 of these points are critical points.

10. Chapter Three: Section One If you are attempting to find the extreme values on a closed interval, you need to do the following;  Evaluate the function at each endpoint  Take the derivative of the function and identify critical values  Evaluate the function at each critical value