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## Critical points

# 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 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.