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so what we want to think about is that 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 for 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 there are all lower values than f of a so this right over here and you can even see it visually this is kind of the classic relative relative maximum value that we've gotten to now what about this so if this was filled in if we were continuous here this would be pretty obviously a relative minimum point but this table does something interesting it jumps up and so this right over here let's say this is the value of f of B that is f of B right over here and 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 less than or equal to f of B so f of B right over here is also 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 a ie is you know your classic relative minimum point but see look at this discontinuity what's going on here but we just have to think about what 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 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 meets it or relative minimum value so that actually is a relative minimum value relative minimum 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 and when X is equal to e this is the function hitting what could really be considered a classic relative 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