Introduction to Relative Minimum and Relative Maximum (What’s the Point?)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn what a relative minimum is and what a relative
maximum is we’ll do lots of examples and you’ll
understand why they’re important let’s do some math [Music]
all right everyone welcome back we’re gonna work through these uh definitions
right here we’re gonna get a good intuitive understanding of what they mean
uh by working out lots of examples and we’re going to understand why these
are important also now before i begin though i want to mention that this
episode is part of the series functions and their graphs step-by-step tutorials
for beginners and so maybe might want to check out some
previous episodes before you get into this one
especially the episode where we talk about increasing and decreasing
functions so we go through that in detail and we talk about those

00:01
um yeah and so today we’re going to talk about relative minimum and relative
maximum now in this video here this episode here i’m
going to be giving you the graphs now in the upcoming episodes we’re going
to be talking about how to find a relative minimum and relative maximum
but we’ll be using a computer and it’ll help us approximate these points
and so i look forward to seeing you in that episode as well all right so let’s
get started so what does it mean right here to be a relative minimum so we got
this um implication down here that we have to understand um and we need to
know um you know about intervals we need to know about functional values
right and so assuming that you’ve looked
at the uh previous episodes let’s try to
move forward now and see what a relative minimum is so let me sketch a graph of
this definition right here so what we have here is a relative minimum and we

00:02
have a function so let’s say we have a function that looks just something like
this right here it’s going like that now intuitively this is a minimum right
here now when i say minimum though i don’t mean like over the whole domain
so this keeps going up this you know just so the domain here would be all real
numbers for this function right here just keeps going keeps going
and so you know this is going to you know come down here so this is not going
to be a minimum globally over the whole domain but it’s minimum
in the sense that it’s decreasing and then it starts to increase
so what we’re going to say is that with respect to an interval let’s call it say
x1 x2 so let’s put down x1 and x2 and this will be our c right here
so a function value c and we have the f of c sitting right here

00:03
so that’s the height right here the f of c is the actual relative minimum
of f so that’s the relative minimum we can see that it’s happening because
there exists an interval and i found the interval here x1 x2 and i don’t really
know what the x1 x2 is i just know that somewhere around here i can find one
and so there’s a c inside of it so that if you choose any other x inside of this
interval the f c will be less than or equal to that x that f of x
so i’ll put the any other x in blue so if i pick any other x say that one
right there then the height here will correspond to something over here f of x
and notice how f c is smaller so that’s f of c is less than or equal to f of x

00:04
so no matter what x’s you choose over here they’re all going to be over here
and the f c is going to be smaller than it so
that’s why we’re going to call this right here the relative minimum
because it’s smaller than all the other output values for any x in this interval
right here now this isn’t true if you expand it out too far
if you expand it out too far and you say your x1 and x2 are out here and here
all right so then maybe it’s still true but what if you extend your x1 all the
way out here and now this is the lowest value here
right and so the important part of this definition is there exists an interval
so it’s so this this implication doesn’t have to hold true for all intervals you
just have to be able to find one and we were able to find one i put it back over

00:05
here this was the x1 this was the x2 so as soon as you can find this interval
right here so that the f c is smaller than all the
other f of x’s no matter what the x is here in this interval here
and then we have a similar definition right here for relative maximum but
rather than go through that argument again let’s start looking at some
examples and i’ll talk about relative maximum as we move along
so let’s look at our first example here all right so our first example is just
going to be a simple line and so i’m going to just say it’s some
line like this right here so let’s say we have the line f of x equals say 2x
and over here we’re going to have something let’s call it a g
and we’ll contrast and compare it with this function right here so this will be
x squared minus 4x and so let’s move that down just a
little bit x squared minus 4x all right and so you know this this function looks

00:06
something like this it goes right it goes to the origin
you know i plug in zero right so it goes to the origin and has this point right
here which is two minus four and so which one of these would you
think has a relative minimum so this right here does not have a minimum at all
and so because no matter uh which y value you pick say you pick minus two i
can find another y value that’s smaller right so this has no uh relative minimum
right here um this one right here does has relative minimum right here so it’s
going to be decreasing and then it starts to increase
and so this right here the height right here minus 4 minus 4 is a relative
minimum relative minimum [Music] of f now why
well i can find an interval right here and let’s say that interval here so

00:07
that’s a 2 right there so i’m going to say 1 3 [Music]
and now any time i pick an x in between 1 and 3 anytime i pick an x
the height here for the function will be less than
or sorry greater than the minus 4 because -4 is the lowest it can go
so -4 is the smallest it’s the minimum and any other x’s i pick along here
we’re going to get some outputs some heights that are
larger than the minus four so minus four is a relative minimum right here
now here we cannot do that if you say you have a relative minimum [Music]
say a one here and then the height here is two
what do you what if you believe that this is a relative minimum here
well then this right here has to satisfy this inequality it has to be less than

00:08
or equal to all of the f of x’s for any x in here but i’ll choose a
number right over here and the height is bigger and i’ll choose an x over here
and the height is smaller so if i choose an x over here this height is smaller
than the 2. so we cannot say 2 is a relative minimum and just this
implication just simply does not hold all right so this one has no relative
extrema this one has one relative extrema we can say -4 is a relative
extrema of f and sometimes we like to use the full sentence -4 is a relative
minimum of f and occurs at x equals and i’ll go small here it occurs at um 2
at x equals 2. so minus 4 is a relative minimum of f and occurs at x equals 2.
so sometimes you’ll need though you’ll need the whole sentence to communicate

00:09
to somebody what’s happening all right so let’s look at another example here
all right so let’s look at a cubic now so now let’s say our function here is
um x to the third and then minus three x squared and then plus two
all right and so let’s see here the graph looks something like this
so we’re going to come up to right here and then we’re going to come down
and then we’re going to go down to about right there
and then we’re going to come back up and then just keeps going up
some people don’t like to put the arrows there it’s just implied but it’s okay
for me um anyways third all right and so this is the point right here 0 2
and this is the point right here 2 minus 2.

00:10
all right good so now we know that from our previous episode we know that it’s
increasing here it’s decreasing here and then it’s increasing here
and so this is going to be a relative maximum and this is going to be a relative
minimum so we can write those sentences out here we can say 2 is a
relative maximum of of f and occurs occurs at x equals zero
and then we can do the same thing for the relative minimum so the relative so
two minus two is a relative minimum it’s always relative to an interval

00:11
relative minimum of f and occurs at x equals 2. all right so there’s the
work there just writing it out but now i want to help you understand
this especially the one for the relative maximum because i haven’t talked about
that definition yet so let’s see why this is a relative
maximum let’s see why this definition is satisfied
and make sure we can parse these words and understand them
so i’m saying the functional value f c which is the output which is the two
i’m saying that two so the functional value two
is called a relative maximum of f if there exists an interval so in order
to justify this statement here i need to actually have an interval
so this is a at 0 right here and so and this is all the way out of 2 right here
so we just need to find an interval right here so i’ll choose a small interval

00:12
right here and if i pick any x in here if i pick any x in the interval
then what will happen so if i pick any x and come up here this
height will be smaller than 2 and if i pick any x in here the height
will be smaller than two of course if we pick zero it’ll be it’ll
be the height will be exactly two so that’s why we have less than or equal to
here or sorry here so this 2 is going to be greater than or equal to
all the other outputs no matter what x you choose
the 2 the output will be greater than or
equal to all the other ones so no matter which x we choose in here it’ll be less
than or equal to the two and so that’s why this is a relative maximum right here
all right and so let’s do uh one more example right here to illustrate this
point right here so let’s let’s get a thing going right here all right

00:13
so let’s look at this function i’ll just call it f again
and it’s going to be here x squared minus 1 with a square root over it
all right and so let’s see what the graph of this looks like
so it’s coming through here and it’s coming through here
and this is the point here minus one and this is the point here one so it’s one
zero and that’s minus one zero right there and so there’s the
sketch there this just continues and continues
all right so there’s the sketch of that right there
and now we’re asking the question are there relative extrema are there any
a graph doesn’t have to have any a graph may have lots of them [Music]
in fact if you if you know what the graphs of sine and
cosine look like you’ll see that perhaps you could have infinitely many relative
extrema but what about this one right here

00:14
so let’s call this one here number four all right so now you may be tempted to
think right here that there’s a relative extrema but the question is is it now
if i say that zero right here which is the height
when we’re right here it’s the height of zero so you may say zero is a relative
minimum because i can find an open interval around here
and if i choose any x’s in here then the height will be greater than
that right there but the problem is is that you can’t actually choose an
interval because there’s no points on the graph to the right of this endpoint
that are close to it i mean there are some points over here
but we need some points directly immediate to it because
we need an interval we need we need to be on the interior of the
domain and we can’t be that so we cannot choose an interval over

00:15
here so there there’s not going to exist an interval on um
this on the other side of this c right here so you would like to say maybe
there is a relative extrema at -1 but then when you start trying to find open
intervals around it you realize there’s nothing adjacent to the -1 right over
here that will give us an open interval that
will give us this interval right here so this is going to have no relative
extrema no relative extrema and we can make the same argument over
here you cannot argue that one is a relative minimum because to be a
relative minimum you would need to have some open interval around that one
but you cannot have an open interval around the one it’s an endpoint
and so that that’s this point right here notice the relative extreme it cannot
occur at an endpoint and the reason why is because any endpoint that you have
you would need to have an interval you need to have an interval so you need
some points to the left and some points to the right

00:16
of the endpoint but an endpoint by definition means it just ends so it’s
not going to have an open interval around it
all right so i hope that helps with this
point right here by looking at something like by some example right here
all right so now let’s go look at another example
so let’s look at this function right here i’ll call this here h of x
and we’ll say it’s x squared plus x plus one all over x plus one
and we looked at a function like this in our previous episode
i’ll show the graph real quick again so it has an isotope at minus one
we’ll put that in here so we’ll say this is x minus one and then it’s going to

00:17
um go like this and actually this has a another isotope here um
and well anyways we don’t we can just specify like that right there
and then we’ll say something like this right here there there’s a
somewhat reasonable graph excuse me all right so this is the point right
here 0 1 and this is the point right here minus 2 minus 3.
and so the question now is the question last time is where is it increasing and
where is it decreasing so last time we said it was increasing on
this interval right here minus infinity to 2 and then union
from 0 to positive infinity and it’s decreasing on

00:18
you know minus 2 right here so here’s the x value for -2 so it’s decreasing
from minus 2 to -1 and then union from -1 to 0. so that’s where it’s decreasing
so we wrote all that down last time check out that video or that episode if
you haven’t seen it but now the question is is this a relative min
and is this a relative max so i’m going to say yes this is a
relative min and that’s because i can find an open interval around it
x1 x2 so here would be my x1 and here would be my x2
and no matter what x i choose in that interval so if i just choose say an x
any one of them and i come up here and the height will be greater than
what’s happening right here at the lowest one right here when we choose zero
that height is the smallest one or perhaps if we chose that x is zero
then we got equals there so here here strength here it’s not

00:19
straight here so in any case yeah i can find an open interval around here where
this is going to be the minimum value so it’s the relative it’s the minimum
relative to some interval so we’re going to say here 1 is a relative minimum
we’ll just say relative min of f at x equals 0.
and then right here is minus two minus three a relative maximum
i say yes it’s not a global maximum right this point is higher than this one
over here but relative to this interval open interval right here
it is a maximum because any x i choose over here any x we choose in this x1 x2
if we go down here and look at a height here
this height is higher this height is greater or perhaps equal and so we got
this inequality right here holding so we’re going to say

00:20
-3 that is the output so -3 is a relative max
so i’ll just abbreviate our rel max relative max of f at
x equals and it occurs at minus 2. all right very good
so there’s some examples there and so let’s see here let’s um
yeah so um that’s good if you have any more questions or would like for me to
do some more examples over relative maximum and relative minimum let me know
in the comments below but also keep in mind that in an upcoming episode i’m
going to show you how to approximate the relative extrema

00:21
using a computer and so that’ll be a lot of fun and so we’ll talk about relative
maximum and relative minimum aka relative extrema in an upcoming episode
and we’ll cover some more examples but let me know how i’m doing and
let me know if you just like this video have a great day and i’ll see you in the
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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