AP®︎ Calculus BC (2017 edition)
Course: AP®︎ Calculus BC (2017 edition) > Unit 6Lesson 2: Justifying properties of functions using the first derivative
- Introduction to minimum and maximum points
- Justification using first derivative
- Calculus-based justification for function increasing
- Justification using first derivative
- Justification using first derivative
- Analyzing a function with its derivative
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?
- 3:25why bigger or equal to? if it's equal how can it be a maximum?(38 votes)
- 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.(19 votes)
- why is it that----- x should be Є of an open interval and not a closed interval(10 votes)
- 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:
- 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?(13 votes)
- 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.(21 votes)
- why do we consider an open interval and not a closed interval to define a relative minimum/maximum point?(11 votes)
- 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.(14 votes)
- 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?(9 votes)
- 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.(7 votes)
- 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)?(5 votes)
- 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!(13 votes)
- 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)?(5 votes)
- The maximum is a number, not a point. So you would say the maximum is 15, and it occurs at x=0 and x=5.(7 votes)
- Are all global minimum and maximum points also relative minimum and maximum points?(5 votes)
- Yes a global maximum is always a local maximum and a global minimum is always a local minimum.(5 votes)
- functions are always a wavy line can they be a straight line?(4 votes)
- All functions of the form y=mx+b are straight lines (where m is the slope and b is the y-intercept).(6 votes)
- What exactly is h? I know it cannot be less than zero, but what exactly is h? What is x E (c-h, c+h)?(3 votes)
- h is a short distance. The interval (c-h, c+h) is a small region around the number c. Specifically, it's an interval of length 2h, centered at c.
We say the point (c, f(c)) is a local maximum if f(c) is larger than every other value in a small region around it.(5 votes)
So right over here I've graphed the function y is equal to f of x. I've graphed over this interval. It looks like it's between 0 and some positive value. And I want to think about the maximum and minimum points on this. So we've already talked a little bit about absolute maximum and absolute minimum points on an interval. And those are pretty obvious. We hit a maximum point right over here, right at the beginning of our interval. It looks like when x is equal to 0, this is the absolute maximum point for the interval. And the absolute minimum point for the interval happens at the other endpoint. So if this a, this is b, the absolute minimum point is f of b. And the absolute maximum point is f of a. And it looks like a is equal to 0. But you're probably thinking, hey, there are other interesting points right over here. This point right over here, it isn't the largest. We're not taking on-- this value right over here is definitely not the largest value. It is definitely not the largest value that the function takes on in that interval. But relative to the other values around it, it seems like a little bit of a hill. It's larger than the other ones. Locally, it looks like a little bit of a maximum. And so that's why this value right over here would be called-- let's say this right over here c. This is c, so this is f of c-- we would call f of c is a relative maximum value. And we're saying relative because obviously the function takes on the other values that are larger than it. But for the x values near c, f of c is larger than all of those. Similarly-- I can never say that word. Similarly, if this point right over here is d, f of d looks like a relative minimum point or a relative minimum value. f of d is a relative minimum or a local minimum value. Once again, over the whole interval, there's definitely points that are lower. And we hit an absolute minimum for the interval at x is equal to b. But this is a relative minimum or a local minimum because it's lower than the-- if we look at the x values around d, the function at those values is higher than when we get to d. So let's think about, it's fine for me to say, well, you're at a relative maximum if you hit a larger value of your function than any of the surrounding values. And you're at a minimum if you're at a smaller value than any of the surrounding areas. But how could we write that mathematically? So here I'll just give you the definition that really is just a more formal way of saying what we just said. So we say that f of c is a relative max, relative maximum value, if f of c is greater than or equal to f of x for all x that-- we could say in a casual way, for all x near c. So we could write it like that. But that's not too rigorous because what does it mean to be near c? And so a more rigorous way of saying it, for all x that's within an open interval of c minus h to c plus h, where h is some value greater than 0. So does that make sense? Well, let's look at it. So let's construct an open interval. So it looks like for all of the x values in-- and you just have to find one open interval. There might be many open intervals where this is true. But if we construct an open interval that looks something like that, so this value right over here is c plus h. That value right over here c minus h. And you see that over that interval, the function at c, f of c is definitely greater than or equal to the value of the function over any other part of that open interval. And so you could imagine-- I encourage you to pause the video, and you could write out what the more formal definition of a relative minimum point would be. Well, we would just write-- let's take d as our relative minimum. We can say that f of d is a relative minimum point if f of d is less than or equal to f of x for all x in an interval, in an open interval, between d minus h and d plus h for h is greater than 0. So you can find an interval here. So let's say this is d plus h. This is d minus h. The function over that interval, f of d is always less than or equal to any of the other values, the f's of all of these other x's in that interval. And that's why we say that it's a relative minimum point. So in everyday language, relative max-- if the function takes on a larger value at c than for the x values around c. And you're at a relative minimum value if the function takes on a lower value at d than for the x values near d.