Section 3 2 F Video Lecture

Hello, Dillon here.

College Algebra students, we are
finally finishing up Section 3.2.

Here we go.

There is one last little topic
we have to talk about.

So, some important vocabulary, especially if you're
going to take more mathematics after this

is the idea of extreme values:
local minimums and maximums.

The technical term is
local maxima, local minima.

These are exactly
what they sound like…

in a very local sense.

And if you focus your attention
on a certain part of the polynomial,

the given point can be
a maximum or minimum.

Here's a value: x equals ‘a.’

The technical way of describing this
in math is a “neighborhood of a.”

What does that mean?

It just means an
open interval around it.

If we just concentrate on this section
of the graph, that is a local maximum.

Why? Because that y value is
greater than everything close to it.

In a similar way, if we concentrate around ‘b,’
again, some small neighborhood around it,

in that open interval around ‘b,’
that is a local minimum.

In this case, the polynomial
that we're looking at

has a local maximum
and a local minimum.

Now, keep in mind, this would be
an odd-degree polynomial.

It doesn't have an overall— what’s called
a global maximum— because this goes forever in—

this y value, as x goes to positive infinity,
y goes to positive infinity.

Do you remember that?

Y goes to positive infinity as
x goes to positive infinity.

This point is a local maximum.

Listen to the vocabulary.

We say: “there is a local
maximum at x equals a.”

The maximum value at that
local maximum is y equals f of a [f(a)].

We say “there's a local minimum at
x equals b”; that minimum value,

that local minimum
value is f of b [f(b)].

All right, and there's
no overall global minimum

because y goes to negative infinity
as x goes to negative infinity.

Here's an important idea.

If we have a
polynomial of degree n,

then the graph of P has at most
n minus 1 local extreme values.

So, the n they're referring to
is the degree of the polynomial.

Remember, we like these
arranged in descending order.

This is the leading term, that determines
the degree of the polynomial.

If it's degree n,
then it can have at most,

one less than that
degree of local extreme values.

“Extrema” is the plural
for extreme values.

Okay, so we're going to finish this up
by just looking at some graphs,

and this is probably screenshots
from a graphing calculator,

but it's good enough
for us to work with.

Notice first of all, the first one here,
the polynomial P-1 of x is fourth degree.

Okay, and here's its graph, and notice
in this case it has no maximum value

because our end behavior is to
positive infinity on both sides.

It does have an overall minimum
value, this local minimum.

It's a little hard to tell what it is,
but it doesn't matter.

What we're just
looking at is the number.

It has a local minimum here,

a local minimum here,

and then a local maximum.

Notice, there are three extrema (that's the
plural for extreme values), and the degree is 4.

Not a coincidence.

Here we have a
fifth-degree polynomial,

that's what the graph would
look like, it’s an odd degree.

Notice this end behavior is consistent with
what we've learned earlier in the section.

Then we have [counting]
1, 2, 3, 4 extreme values.

Notice that's one
less than the degree.

The theorem up here says that it can
have at most n minus 1 local extrema.

That doesn't mean it
has to have them, okay?

Now in these two examples,
it matched exactly:

fourth degree/three extrema,

fifth degree/four extrema.

The point is it can't
have more than four.

Here's exactly what
I'm talking about.

In this last example, so look: fourth degree,
it can have at most three (n-1)—

three extreme values.

But in fact, this
polynomial only has one.

That's still consistent with the
theorem that we just saw.

This just helps us—

it’s an additional tool for us to use
when we're sketching a graph.

Okay, wow, that's it.

We finally finished Section 3.2.

On to 3.3, bye.