Tip:
Highlight text to annotate it
X
But Taylor series.
That explains.
And that's the whole reason to [...] work the second derivative test now in terms of Taylor series.
Because the Taylor series way of describing the second derivative test ...
... will tell you exactly what to be next.
And exactly what to do next, and then next, and then next.
And eventually it will tell you everything.
So let's look at that.
Er ...
This is Taylor series for our function at the critical point x zero.
What do you know about this Taylor series at that critical point?
What does it mean that you have x zero being critical point?
It means derivative is zero, right?
It means that first derivative is zero.
That's exactly what it means to be a critical point.
So at critical point you have that term being zero.
And the Taylor series looks like a number ...
... plus ... So we skip the linear term and we come with quadratic.
Times x minus x zero squared.
Now if we stop there ...
... and we say instead of looking at the function itself ...
... I will look at the approximation of that function ...
... by that quadratic polynomial ...
Can you tell me if this ...
... parabola. Isn't it a parabola?
Can you look at any parabola algebraically and tell me if it has minimum or maximum?
Let me give you three minus five x plus six x squared. Does it have minimum or maximum?
Is it like that or is it like this?
Student: Yeah. NB: Right? How do you know?
Student: [...].
That coefficient is positive.
All you have to know ...
... to decide if it has minimum or maximum is the coefficient with x squared.
So you completely decide about this parabola by looking at this term.
And deciding well positive or negative.
If positive then minimum.
If negative ...
... then maximum.
And of course how do you decide about that number?
We just look at numerator.
And that's what at the end of the day the second derivative test tells you.
Look at that number -- second derivative at x zero ...
... and check if it is positive or negative.
It never explains the reason why you do that.
Or sometimes it explains the reason.
Sometimes people explain that ...
... the second derivative of a function ...
... is about how the ...
... graph behaves if you look at the tangent lines.
If the second derivative is positive that means ...
... the first derivative increases.
And being zero ...
... at the critical point increase means ...
... going this way.
And being zero ...
... at the critical point decrease means ...
... going that way.
So there is a possible geometric explanation.
Now the point is that even having that geometric explanation ...
... you wouldn't know what to do next.
If your second derivative is zero.
Right? What is it that you look at?
If the second derivative is zero ...
... and we don't know if it goes this way or that way.
So you noticed last time that geometry fails to go beyond degree two.
Right? Approximately [...] things.
Well it is my believe that we just fail to see ...
... beyond degree two geometrically.
You sort of see degree two but nothing beyond that.
And ...
Algebraically we can go as far as we wish.
So if the second derivative is equal to zero.
What does that mean?
Algebraically it means that not only this term vanishes, but that term vanishes.
So any suggestions, what to do?
Student: [...].
Take the next term. Right?
So if ...
... the second derivative at x zero is zero ...
... then the function ...
... is approximately ...
... still that number plus the third term.
The third derivative at x zero ...
... divided by three factorial times x minus x zero cubed.
And then what? How does this look like?
A cubic function.
Does it have minimum or maximum?
A function like y equals x cubed.
Does it have minimum or maximum?
Student: Neither. NB: Neither.
You know that immediately.
So if you look at this ...
... and you see that this term is not zero ...
Well if not zero ...
... then no minimum, no maximum.
It's probably something like reflection point ... well inflection point, right?
And if zero ...
... then this third derivative test fails.
But do you know what to do next?
Go for the next term, right?
So if ...
If also the fourth derivative ...
... well the third derivative of x zero is zero ...
... then ...
... the function is that number plus the fourth derivative at x zero over ...
... four factorial times x minus x zero to the power four.
So the expression like x to the power four.
Does it have minimum or maximum?
y equals x to the power four has minimum. You know that, right?
But the function like minus two x to power four has maximum.
And that means it depends only on this number.
So you look at this number.
You get number is positive.
That minimum.
If negative ...
... then maximum.
And there is slight possibility ...
... that this number is ...
... zero.
And the fourth derivative test fails.
But you still know how to go further.
All these tests.
Right? So Taylor series provides you with the whole sequence of tests.
Like that you can keep doing trying to decide what happens at this critical point.
Now the fundamental question is ...
Can you fail at the end?
What if you try the second derivative test, it fails.
You try the third -- it fails. You try the fourth -- it fails.
You try and try and the whole life trying and you fail every time.
Is that possible?
NB: But what does it mean? Student: It's a line.
It means that all these coefficients are zeros.
And that means the function is equal to a number, a constant.
So if you fail all the time ...
... that's a constant.
So you see Taylor series tells you exactly what to do.
And you essentially never fail.
If you know all derivatives are going to be zero you know exactly what the function is.
Not only at that point but everywhere.