Points of Inflection

As MEI has just had its new A-Level Maths specification accredited, I thought I would glance back over their sample assessment materials. On one of the papers for the full A-Level, I came across this question:

Now I stared at this question for a while. I hope I'm not the only one, but if I am I'm willing to take the hit.

Either because my A-Level Maths teachers didn't realise, or they didn't teach it because it was not something that would be tested, I was led to believe that points of inflection were just types of stationary points, where the curve has either positive or negative gradient on both sides of the stationary point. A classic example is y = x^3 with its stationary point at the origin.

Checking with my colleagues and researching the actual definition online, it turns out that points of inflection on a curve are where the curve changes from being concave to convex (or vice versa), and this does not have to be a stationary point.

So the two points I have circled on the graph is where these points of inflection appear.

To find them, you have to find the second derivative of the curve and find where this is equal to zero, giving you in this case two values of x (the x-coordinates of the points of inflection).

I have certainly never seen a question like this on an exam paper before, and this could easily have sneaked past me if I hadn't read through the questions on the specimen paper carefully. I then looked up points of inflection in the content statements and found:

So it is there, and can be examined. I'm happy with that - I've learnt some new maths and another use for the second derivative, other than for checking whether a stationary point is a local maximum or a local minimum. It always made me think the second derivative was a bit useless, because if it's zero at a stationary point you've learnt nothing. Why not just look at the gradient either side of the stationary point?

Then I kept looking. One more search of the word inflection through the content statements found this:

Now what had points of inflection to do with the normal distribution? I took a moment to think about the shape of the bell curve and clearly it has two points of inflection, one either side of the mean and the centre, but what relevance does this have to the distribution?

So here's the equation of the normal distribution curve:

Now we can't integrate this algebraically to find areas - that's why we use statistical tables and calculators. But to find the points of inflection, we need to differentiate. In fact, we need to differentiate twice to find the second derivative. This we can do, using a combination of the Chain Rule and the Product Rule.

It's not easy, however, and I made an initial mistake with my Chain Rule as you may be able to spot:

But it's a great result at the end and definitely worth the effort. It turns out that the points of inflection of a normal distribution curve appear a single standard deviation away either side of the mean.

This is definitely an extension problem I will set to my second years in the linear A-Level. The problem can be differentiated and made more approachable if you focus your students' attention on the standard normal distribution, with the mean = 0 and the standard deviation = 1.

If anyone has any other ways of extending this problem and/or further problems involving points of inflection that aren't stationary points, I would be very interested.