Three panels, one x, and a test that is faster rather than stronger.
Move to x = 1 slowly and watch the colour change.
The top curve is coloured by the sign of f″, not f′. At x = 1 the colour flips at exactly the moment the bottom graph crosses zero, and the curve is still rising throughout.
That is the point students miss: concavity is a completely separate question from direction, and a curve can be going up the whole time while changing its mind about how.
If f″ is zero at a stationary point the test says nothing at all, and you fall back to a sign check of f′.
y = x⁴ at the origin is the example to have ready: f′ and f″ are both zero and it is still a minimum.
| The function | f(x) = x³ − 3x² + 1, f′ = 3x² − 6x, f″ = 6x − 6. |
| Stationary points | x = 0 and x = 2. f″(0) = −6 so (0, 1) is a maximum; f″(2) = 6 so (2, −3) is a minimum. |
| Point of inflexion | f″ = 0 at x = 1, and the concavity genuinely changes, so it is (1, −1). |
| 1. f″(2) | 6. |
| 2. f″ negative at a stationary point | B, a local maximum. |
| 3. y of the inflexion | −1. |
1 markDifferentiating twice correctly.
1 markStating the SIGN of f″ and the conclusion that follows. The sign alone is not an answer.
1 markFor an inflexion, solving f″ = 0 and substituting back into f for the coordinate.
1 markConfirming the concavity actually changes, when the question asks you to justify.
| They give | What it means |
|---|---|
| 0 (Q1) | Gave f′(2), which is zero because x = 2 is stationary. They differentiated once and stopped. |
| −3 (Q1) | Gave f(2), the y coordinate. Three similar-looking numbers are in play and the labels matter. |
| Minimum (Q2) | Sign confusion. Concave down is a cap, so the stationary point sits at the top. Draw both shapes rather than restating the rule. |
| 1 (Q3) | The x coordinate again. Same habit as Standard Level 5.6, and it does not go away on its own. |
| 0 (Q3) | Substituted into f″, which is zero there by definition. |
"Is a point of inflexion always a stationary point?" No, and this example proves it: at (1, −1) the gradient is −3, nowhere near zero. Worth settling early because the two ideas get fused.
"Concave up or convex?" The guide uses concave-up and concave-down, so use those. Convex is correct English and will cost nothing, but consistency with the exam wording is free.
"Why bother with the sign table then?" Because the second derivative test fails when f″ = 0, and because some questions ask for it explicitly. It is a shortcut, not a replacement.
| What is happening | |
|---|---|
| 1 | Recall 5.2: what f′ tells you. Then ask what f′ does NOT tell you, and let the gap open. |
| 2 | The three panels. Sweep slowly, stopping at both stationary points and the inflexion. |
| 3 | The second derivative test on the worked example, both stationary points. |
| 4 | Points of inflexion, including the x⁴ counterexample where the test fails. |
| 5 | Questions 1 to 3. |
| 6 | Notation: f″(x) and d²y/dx² are the same thing. |
Do not say "f double dash is zero means a point of inflexion". It is the same false shortcut as "f dash is zero means a turning point", one level up, and it is just as wrong.
Do not drop the sign check entirely now that the test exists. The first time f″ comes out zero at a stationary point, a class that has only ever used the test has nothing to fall back on.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
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