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Remember that ...
Second derivative test.
Do you remember even what it was for?
No?
So that was about ...
... studying a function.
And what you can figure out about a function is ...
Well basically all the Calculus I is about ...
... studying a function from the point of view 'find maximum of function, find minimum of function.'
That's it.
Essentially you study for the whole semester to solve that simple problem.
That given ...
... graph as a picture you can solve momentarily.
Right?
Bum! That's maximum. And that's minimum.
It is only that algebraic descriptions without a picture ...
... make your work hardened do some manipulations before you figure out where the maximum is, where the minimum is.
So how do you figure out maximum or minimum of a function?
Well of course you differentiate, right?
So you take first derivative.
And then you equate it to zero.
And then what do you do?
You solve, right? So solve and ...
Solve for x.
And what you find has a name.
Those are critical points.
So you find critical points.
And what if you found several of them?
How do you decide if the point you have is maximum?
Or what if you found one point?
How do you decide whether it is maximum or minimum?
Student: Take second derivative and see if it is positive or negative.
Ah! That's the second derivative test.
So you take second derivative.
So find critical points.
Let's call it x zero.
And then look at the second derivative ...
... at the x zero.
And see if it is positive or negative.
And then if ... so ...
If the second derivative is positive then what?
Then minimum.
If the second derivative is negative.
Then you have maximum.
And what if the second derivative is ...
NB: ... zero? Student: You take [...].
You take third derivative?
Ah, no, no, no!
This is second derivative test.
It stops there. Right? So ...
The test fails.
Right? This is what standard Calculus I course tells you.
Now the curious thing is that there are some examples of very nice functions.
So let's look for example.
At a function f of x equals x to power four.
Isn't that a simple function?
Do you know if it is ... if it has minimum or maximum?
Do you know how the graph looks like?
So does it have minimum or maximum?
Student: Minimum.
Well of course minimum. You see it immediately, right?
Now what if you apply the second derivative test?
The first derivative is four x cubed.
If you equate that to zero the only solution is x zero equals zero.
This is the only critical point.
And then you compute the second derivative.
Which is four times three x squared.
And then you evaluate ...
... the second derivative at the critical point.
And that is ...
And second derivative test doesn't work even for that simple function.
And of course it will not work for x cubed, it will not work for x to the power six, seven.
It doesn't work for any power greater than two.
So if it doesn't work for simple polynomials like that ...
... even in cases when you clearly know what the answer is ...
... well how good a test is that?
So it's [...] test. I'm not complaining too much.
But I'm just asking 'well if it fails in these cases what should I do?'
And that's what Calculus I doesn't tell me.
Right? Because there is no third derivative test explained there.
There is no fourth derivative test explained there.