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.