In order to calculate points of curvature, so we’re talking about maximum, minimum, or points
of inflection, we now need to use the second derivative. Now, we’ve done the second derivative
a few times through this course but without really talking about why we’re doing it. This is really
important because this tells us what happens to the curve after the stationary point. So whether
after the stationary point the curve goes up or whether after the stationary point the curve
goes down or whether it continues with the same gradient, it’s the second derivative that allows
us to find that. We often say it measures how concave a curve is. Let me just show you what I mean.
In a minimum point, you would have a point like this. So let’s just call this our stationary point.
After the point, it goes up in order for this to be a minimum. This means that the second derivative
here, d²y/dx² has got to be positive because remember, this is a positive gradient. So anything
that goes upward is positive. Anything that goes down is a negative gradient. In this case,
d²y/dx² is greater than or equal to zero because it’s positive. This kind of curve, if I draw
it properly, gives us a minimum curve as we’ve just discussed on the previous slide. Similarly, if you
have a stationary point and straight after the stationary point, the curve goes downward, we see
a negative gradient. We can say here that the second differential, d²y/dx² must now be
less than zero. We call this point a maximum point as we’ve just discussed. These are just little
visual features that we’ll have to familiarize ourselves with. With the point of inflection, we have
already said that we have a flat point. So it could be positive and positive gradients, remember
we’re talking about and it could be flat and negative and negative. That should be a little bit
more curvy because we’re talking about curves. So this here is a point of inflection.
Now, let’s just discuss how we do this numerically. You’ve found the gradient by doing dy/dx of a function.
You equal it to zero because remember that at stationary points, so I’m just going to shorten this,
at stationary points, dy/dx equals to zero as we’ve stated earlier. So you’ve found your stationary
points or the coordinates of the stationary points by equaling it to zero. You then want to know
what kind of stationary point it is. All you do, you take your dy/dx and you differentiate it
the second time. That gives you your second differential here. Once you know your second
differential, you substitute your x or y values in there or both and then you look at whether
you get a positive answer, a negative answer, or a zero. In this point here, in our points of
inflection, these are the only points where we say that d²y/dx² equals to zero.
Then this requires a little bit more investigation. We investigate by using gradients before and after
the point. Let me just highlight this for you here. We are saying that if we have a second derivative
greater than zero, you will have a minimum point. If the second derivative is less than zero, you
will have a maximum point. If the second derivative is equal to zero, we have a point of inflection
which we will investigate a little bit further. I’ll show you how that’s done in an example.
So just to summarize, stationary points can be found by doing dy/dx equals to zero.
You’ll see that we’ll do so much of this that this will just be something that you’ll manage to learn.
But the second part of it which is important is finding out the nature of the stationary points,
what type of stationary point this is, we use the second differential. Now, in order to find the second
differential, we say that the derivative or the second derivative must be greater than or equal to zero.
If it is greater than or equal to zero, you get a minimum point. If the second differential is less than
or equal to zero, you get a negative answer, so that gives you a maximum point. If the second
differential is equal to zero, we’ll deal with this shortly, but we call this a point of inflection.
Now, this can often get quite confusing. I like to think of it as the second derivative being positive.
When you’re positive, you can imagine that that’s a smiley face. So, a smiley face looks like
a minimum point when you sketch it out. Second differential being negative is obviously, you can
think about being an upside down curve, a negative face or a negative smile. So that gives you
a maximum point. And a point of inflection is easy to remember, so that equals to zero.
You will have curves which you’re dealing with which have positives and negatives
and maybe a point of inflection at the same time. You have to conclude or analyze those graphs
or those values fairly quickly. So it’s important to find some faster ways of working that out.