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# Introduction to minimum and maximum points

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.2 (EK)
CCSS.Math:

## Video transcript

so right over here I've graphed the function y is equal to f of X I've graphed it over this interval looks like it's between zero 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 them a maximum point right over here right at the beginning of our interval 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 is a this is B the absolute minimum point is f of B 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 kind of 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 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 this value right over let's call this let's say this right over here see this is C so this is f of C we would call f of C is a relative relative maximum value relative maximum value and we're saying relative because it's obviously the function takes on 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 F of D is a relative minimum point or relative minimum value F of D is a relative minimum or a local minimum value once again there's in the over the whole interval there's definitely points that are lower and we hit an absolute minimum for the interval at is equal to B but this is a relative minimum or a local minimum because it's lower than B as 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 you know it's it's fine for me to say well you're in a relative maximum if you're at a larger if you hit a larger value of your function than any of the surrounding values and you're in 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 we say that F of C F of C is a relative relative max relative maximum value if if F of C is greater than or equal to f of X for all X for all X that we could say we could just say kind of in a casual way for all X near 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 that's within an open interval of C - H - C plus h where H is some value greater than 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 and you 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 + H that value right over here C minus H and you see that over that interval 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 what the more formal definition of a relative minimum point would 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 zero so you can find an interval here so this 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 is of all of these other X's in that interval and that's why we say that's a relative minimum point so two everyday language relative max if you're larger than the then the if the function takes on a larger value with the value at C then for the X values around C and your relative minimum value if the Valentich saan a lower value at D then for the x values near D