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Identifying relative minimum and maximum values
So what we want to think about is, at what x values does our function here in orange-- let me make this clear. This is the graph of y is equal to f of x. At what x values, and we have some choices here, which of these x values I should say, does f of x hit relative maximum values or relative minimum values? And I encourage you to pause the video and think about it and classify whether we hit a relative maximum value or a relative minimum value at each of these x's. So let's first look at x equals a. f of a is right over here. This is f of a. And I can pretty easily construct an open interval around a so that f of x, if x is in this open interval, is going to be, is definitely going to be less than or equal to f of a. f of x in that interval is definitely, they're all lower values of f of a. So this right over here, and you can even see it visually, this is kind of the classic relative maximum value that we've gotten to. Now what about this? If this was filled in, if we were continuous here, this would be pretty obviously a relative minimum point. But this does something interesting, it jumps up. And so this right over here, let's see, this is the value of f of b. That is f of b right over here. This is a little bit counterintuitive. But I actually can construct an open interval around b. I can actually construct an open interval around b where the value of f of x, if it's in that interval, is a less than or equal to f of b. So f of b right over here is also a relative maximum value. Now what about c right over here? Well if this was just at the bottom of a kind of, if it would look like e, e is your classic relative minimum point. But c, look at this discontinuity. What's going on here? But we just have to think about, well can we construct an open interval around c where f of c is-- this is f of c right over here-- where f of c is less than or equal to the x's in, is less than or equal to f of x for the x's in that open interval. Well let's see, in this open interval the way I've drawn it, the f of x's are here and they are over here. So it looks like f of x is always greater than or equal to f of c. So that by that definition, by the definition of a relative minimum point, this makes it, or relative minimum value. So that actually is a relative minimum value. Now we get over here to d. And really, by the same argument that we used for b, that is also at d our function takes on another relative maximum point. And then e, when x is equal to e, this is the function hitting what could really be considered a classic relative minimum point. We can easily construct an interval where you take any x in that interval, f of x is going to be greater than or equal to f of e. So this is a relative minimum value as well.