Relative (local) extrema
Introduction to minimum and maximum points
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.