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## Critical points

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# Identifying relative minimum and maximum values

## Video transcript

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.