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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 5

Lesson 4: Using the first derivative test to find relative (local) extrema

# Introduction to minimum and maximum points

Sal explains all about minimum and maximum points, both absolute and relative. Created by Sal Khan.

## Want to join the conversation?

• why bigger or equal to? if it's equal how can it be a maximum? • f(x) refers to any point on the "line" or on the system of solutions, since x is a variable meaning that since x varies it could be equal to the value c. So if Sal wrote greater instead, it would mean that he was saying that it was possible for c to be greater than c or if c was equal to 8, that 8 was greater than 8. See, it doesn't make any sense.
• why is it that----- x should be Є of an open interval and not a closed interval •  Great question, and a deep one too.

Sal writes x ε (c - h, c + h) to describe the open interval around c. What is meant is that, c will be contained somewhere inside the interval and will not be either of the endpoints, written as a < c < b.

Now, here is the deep part. Soon, if you continue on in math, you will learn in calculus how to use the process of differentiation to find maxima and minima. The thing with differentiation is it is based on the concept of the limit and that process requires that all points subject to differentiation be interior points in the set, which is only possible with open sets of the form x ε (c - h, c + h) or a < c < b.

This open interval, or neighborhood, or open n-ball concept figures prominently in more advanced math in many areas such as abstract algebra, calculus, topology and many others.

For now, accept that there is a very good reason the interval is open. When you have more math experience, in a year or two, you may have an "aha moment" if you remember your question in the face of the technical explanation.

Meanwhile, for fun:
https://www.mathsisfun.com/calculus/limits.html
https://www.mathsisfun.com/calculus/maxima-minima.html
• The only question I have is why do we care? What does it imply and where and how would we use it in math or other fields? • One application is business & finance: finding the maximum on an xy graph of profits & expenses can locate the sweet spot where profits are maximized and expenses are minimized.

Maximum and minimum points are also used by biologists and environmental scientists to project things like maximum population growth for different species.
• why do we consider an open interval and not a closed interval to define a relative minimum/maximum point? • Good question. It would seem that if there is an open interval that satisfies the condition, there must be a closed interval that satisfies the condition, simply by taking two points within the open interval on either side of the local min/max. Unless I'm missing something, either type of interval would do just as well.
• If a function plateaus (I know there are better examples, but square waves are the first that come to mind), would all of those points on the horizontal section be considered relative maxima or minima? • The points within a horizontal interval (but not the endpoints of that interval) are considered to be BOTH relative maxima and relative minima at the same time. However, the endpoints of the interval that is horizontal would be considered only a max or min, depending on what the function does outside the horizontal interval.

A point on a curve is considered a relative maximum if the function is defined at that point and the function has equal or lower values an infinitesimal distance on both sides of the point. If the point is the endpoint of a CLOSED (not open) interval, then the function need only be equal or lower than than that point on the side that lies within the interval.

A point on a curve is considered a relative minimum if the function is defined at that point and the function has equal or greater values an infinitesimal distance on both sides of the point. If the point is the endpoint of a CLOSED (not open) interval, then the function need only be equal or greater than than that point on the side that lies within the interval.

Thus, the endpoints of a CLOSED interval are always considered to be relative maxima or relative minima (note the term for a point that is either a max or min is "extremum" which has the plural "extrema"). However, the endpoints of an OPEN interval are never considered to be extrema because they do not lie within the interval.
• First of all, how are you to decide where the "h" is? second, who is to decide the value of h? and third, why do we say c-h and c+h; if "h" is another point on the line then c+h would be greater than f(c)? • h is just some number greater than 0. The value of h isn't established. The reason why is that for some open bound (c-h, c+h), f(c) is greater than or less than all f(x) in that interval.

You have to realize if h>0, then c-h is the x-value of some point to the left of c, and c+h is the x-value of some point to the right of c. Thus, the open bound (c-h, c+h) includes all x strictly between those two points. And if we say f(c) is a relative minima, we are really saying f(c) is less than all f(x) in the interval (c-h, c+h). Similarly, if we say f(c) is a relative maxima, we are really saying f(c) is greater than all f(x) in the interval (c-h, c+h).

To summarize, h is just some positive number that helps establish an open bound to the left and right of c. We can use c-h and c+h to establish the definitions of the relative maxima and minima.

Hope this helps!
• What if you have two coordinates on a graph that have the same highest output out of all the other outputs defined in the function. How would I represent the two maximums? Would I do say, (0, 15) U (5, 15)? • Are all global minimum and maximum points also relative minimum and maximum points?   