Let’s have a look at our first example. We are being asked to find stationary points.
When we say stationary points, we’re finding the coordinates of the stationary points when y = x².
Let’s try it out using all the theory that we’ve just looked at and see if we can actually find out
what kind of curve this is and where the stationary points are. Our curve is y = x².
Now you should roughly have an idea of what this looks like graphically anyway. But even if you don’t,
with the calculations we’re about to do, you’ll be able to figure it out precisely. So we know that
this is some type of a curve. If we want to find its stationary point, we want to say that at stationary
points, remember the theory that we’ve just discussed, dy/dx = 0, you can say the gradient
is zero but to make it more mathematical, we are obviously dealing with the differential dy/dx.
So we simply differentiate this equation, so y = x². We do dy/dx. X², bring the power down,
decrease the power by 1, goes to 2x. From here, remember we need to find the stationary points
by equaling the gradient to zero. So we’re just going to equal this to zero. So essentially, we just
have to solve 2x = 0 which then gives you x = 0. So now, it’s an easy calculation here.
What we’ve just found is that if I have a graph XY, I have a stationary point when x = 0.
Now, in order to find that out on this graph, we also need to know the y value. So we can find y
by substituting x = 0 into the original equation, into y = x² because that’s at y equals to something.
To find y, we can just substitute x = 0 there. So y = 0² also gives you 0. Quite fortunate this has turned out
to be so nice and straightforward. So we’re saying that on this graph, at (0, 0) when y = 0 and x = 0,
we have a flat point here. So we have a stationary point. That’s all we know about this graph so far.
But we also want to know what kind of stationary point this is. So it’s no use knowing that the graph
goes flat. It must be something else going on as well. In order to find that out, we can now do
the second differential. So we can say, "Let's find the nature of this stationary point."
Remember what I said about the nature? In order to calculate the nature, you need to do
the second differential. We’re now, if I just bring out my dy/dx which we’ve worked out already is 2x,
I now need to work out the second differential. So I’m just going to differentiate this again.
D²y/dx², differentiate 2x, that just gives you an answer of 2. This is a fairly simple example.
So we don’t really have to substitute any x values because this x has just disappeared. In the examples to come,
you’ll see that the x’s will usually stay and then we’ll substitute it in to find what the nature is.
But anyway, let’s talk about this number 2 here. Two is a positive number. So you can say that that
is greater than zero. So we’re saying that the second differential is greater than zero. Think about what
that means. So go through your three definitions of second differential. What happens
when it’s greater than zero, less than zero or equal to zero. Greater than zero means it’s positive.
So I’m just going to imagine positive being a bit of a smiley face there but you can also then analyze
that this means that it is a minimum value. So now, I can put that back in here.
So, this stationary point is in fact the stationary point that looks like that. You can then extend this graph
because there’s nothing else going on. So basically, you’ve got an x² curve. For the ones who knew
what y = x² looks like, you know that it’s just this parabola, a positive parabola.
For the ones who didn't, you've just done it from scratch by doing stationary points
and calculating the nature of the stationary points.