Identifying absolute or global extrema

- [Voiceover] So we have the function

g of x is equal to x
squared times the natural

log of x.

And what I wanna do in this video is see

if you can figure out the absolute extrema

for g of x.

So are there x values where
g takes on an absolute

maximum value, or an
absolute minimum value.

Sometimes you might call
them a global maximum,

or a global minimum.

So the first thing I like
to think about is well,

what's the domain for which
g is actually defined?

And we know that in the natural log of x

the input into natural log, it
has to be greater than zero.

So the domain, the domain
is all real numbers

greater than zero.

So x has to be greater than zero.

Anything lateral log
of zero is not defined,

there is no power that you could take e to

to get to zero, and natural
log of negative numbers

is not defined.

So that is the domain.

The domain is all real numbers such that

all real numbers access so
that x is greater than zero.

So our absolute extrema have
to be within that domain.

So to find these, let's
see if we can find some

local extrema and see if any
of them are good candidates

for absolute extrema.

And we could find our local extrema

by looking at critical
points, or critical values.

So let's take the derivative of g.

So g prime, using a new
color just for kicks.

All right, so g prime of x is equal to,

we could use the product rule here.

So derivative of x squared which is two x

times the natural log of x

plus x squared times the derivative

of natural log of x,
so that is one over x,

and I can just rewrite that.

X squared times one over x.

And we're gonna assume that x is positive.

So that is going to be,

that is going to be just,

and actually I didn't even
have to make that assumption

for what I'm about to do,

x squared divided by x
is just going to be x.

All right.

And so that is g prime.

So now let's think about
the critical points.

Critical points are where the derivative

they're points in the domain,

so they're gonna have to
satisfy x is greater than zero,

such that g prime is either undefined

or it is equal to zero.

So let's first think about
when g prime is equal to zero.

So let's set it equal to zero.

Two x natural log of x
plus x is equal to zero.

Well we can subtract, we could
subtract x from both sides

of that and so we get
two x natural log of x

is equal to negative x.

See, if we divide both sides by x

and we can do that, we
know x isn't gonna be zero,

our domain is x is greater than zero.

So this is going to be,
actually let's divide

both sides by two x.

So that we get the natural
log of x is equal to

negative one half.

Negative x divided by two
x is negative one half.

Or we could say that x is equal to

e to the negative one half

is equal to x.

Remember, natural log is just log base e.

So e to the negative one half,

which we could also write like that,

e to the negative one half,

or one over the square root of e.

So that's a point at which g,

at which our derivative I should say

is equal to zero, it is a
critical point or critical value

for our original function g.

So and that's the only place
where g prime is equal to zero.

Are there any other points
where g prime is undefined?

And they're have to be
points within the domain.

So let's see, what would
make this undefined?

The two x and the x, that
you can evaluate for any x.

Natural log of x, once again,
is only going to be defined

for x greater than zero.

But that's, we've already
restricted ourselves

to that domain, so within the domain

any point in the domain our derivative

is actually going to be defined.

So given that let's see what's
happening on either side

of this critical point.

On either side of this critical point.

And I could draw a little
number line here to

really help us visualize this.

So, if this is negative one, this is zero,

this is let's see, e to
the, this is gonna be like

one over, oh boy this is,

this is going to be a
little bit less than one,

so let's see, why don't we plot one here,

and then two here, and so
we have a critical point

at one over the square root of e,

and we'll put it right over there.

One over the square root of e.

And we know that we're only defined from,

for all x's greater than zero.

So let's think about the interval between

zero and this critical point.

Right over here.

So the open interval,
from zero to one over

the square root of e,
let's think about whether

g prime is positive or negative there.

And then let's think about it for

x greater than one over
the square root of e.

So that's the interval from
one over square root of e

to infinity.

So over that yellow interval,
we let's just try out

a value that is in there.

So let's just try g
prime of, I don't know,

let's try g prime of 0.1.

G prime of 0.1 is definitely
going to be in this interval.

And so it's going to
be equal to two point,

two times 0.1 is equal to,

is equal to 0.2 times the
natural log of 0.1 plus 0.1.

And let's see.

This right over here, this is
going to be a negative value,

in fact it's going to be quite,

it's definitely going to be
greater than negative one.

Cuz e to the negative
one only gets you to,

let's see, e to the
negative one is one over e

so that's one over 2.7, so
one over 2.7 is going to be,

so this is going to be around 0.3 or 0.4.

So in order to get point one,

so this'll be around 0.3 to 0.4.

So in order to get to 0.1 you
have to be even more negative.

So this is going to be, I could
say less than negative one.

So if this is less than negative one,

and I'm multiplying it times 0.2,

I'm gonna get a negative
value that is less than,

less than negative 0.2, and
if I'm adding 0.1 to it,

well, I'm still going
to get a negative value.

So for this yellow interval, g
prime of x is less than zero.

And it would be, I should
have gotten a calculator,

or I could have gotten a calculator out,

I could have just evaluated a lot easier.

So g prime of x is less
than zero in this interval.

Now let's see in this blue
interval what's going on.

And this'll be easier,
we could just try out

the value one.

So g prime of one is equal
to two times the natural log

of one plus one.

Natural log of one is just zero.

So all of this just simplifies to one.

So over this blue interval,
I sampled a point there,

g prime of x is greater than zero.

So it looks like our
function is decreasing from

zero to one over square root of e,

and then we increase after that.

And we increase for all
x's that are greater than

one over the square root of e.

And so our function is going to hit,

if we're decreasing into that and then

increasing after that, we're
hitting a global minimum point,

or a absolute minimum
point at x equals one

over the square root of e.

So let me write this down.

We hit a, we hit a absolute minimum

at x equals one over

the square root of e, and
there is no absolute maximum.

As we get above one over
the square root of e

we are just going to think about

what's going to be happening here.

We're just going to one,

we know that our function
just keeps on increasing

and increasing and increasing forever.

And you could look at even this,

x squared is just gonna get
unbounded towards infinity,

and natural log of x is gonna
grow slower than x squared,

but it's still gonna go
unbounded towards infinity.

So there's no global,
or no absolute maximum.

No absolute maximum point.

And now let's look at the graph of this

to feel good about what
we just did analytically,

without looking at it graphically.

And I looked at it ahead of time.

So let me copy and paste it.

And so this is the graph of our function.

So as can see when this
point right over here,

this is when, this is one
over the square root of e,

it's not obvious from looking at it

that it's that point.

X equals one over the square root of e.

And we can see that it is
indeed an absolute minimum point

here and there is no
absolute maximum point.

There's arbitrarily high values

that our function can take on.