What f′ and f″ look like underneath f, and how to tell which is which when nobody labels them.
Here are f, f′ and f″ for the same function. Move the line and watch what happens at the same x in all three. Then hide the labels and see whether you could still tell them apart, because that is what the question usually asks.
Watch x = 1, x = 2 and x = 3. Something happens at each one, in a different graph each time.
For f(x) = x³ − 6x² + 9x:
| at x = | 1 | 2 | 3 |
|---|---|---|---|
| f | maximum, 4 | inflexion, 2 | minimum, 0 |
| f′ | crosses zero | minimum, −3 | crosses zero |
| f″ | −6 | crosses zero | 6 |
The inflexion at x = 2 is not a turning point. The gradient there is −3, nowhere near zero. f″ is zero, so the bend changes, but the curve carries on downwards throughout. That is a point of inflexion with a non-zero gradient, and it is the case most often missed.
A very common question gives you the graph of f′ and asks about f. Work from the sign and the zeros:
| If f′ is | then f is |
|---|---|
| above the axis | increasing |
| crossing the axis downwards | at a maximum |
| crossing the axis upwards | at a minimum |
| at its own minimum or maximum | at a point of inflexion |
| touching the axis without crossing | at a stationary point of inflexion |
1. For f(x) = x³ − 6x² + 9x, find f″(1).
2. The graph of f′ has a minimum at x = 2. What is happening to f at x = 2?
3. Give the y coordinate of the point of inflexion.
When sketching f′ from f, get the zeros in the right places first. Everything else follows, and the zeros are where the marks are.
Say which feature of which graph you are using. "f′ has a minimum at x = 2, so f has a point of inflexion there" is a complete reason; pointing at a picture is not.
Degrees drop by one each time. If you sketch f′ of a cubic as another cubic, something has gone wrong before you started.
These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.
Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.
Get a free verdict Full written review, $99
I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.