Differentiate
Find f′(x) for the cubic.
AQA 8365 · 4.3/4.6/4.7 · Stationary points
Stationary points: classifying with the second derivative
The classification mark, and the y-coordinates. Solving f′(x) = 0 gives x-values only; the marks continue through substitution and the second-derivative test.
Reviewed family · Studara-authored items · No grade-uplift claim
The lost mark
One answer. Several places the method can fail.
Studara does not reduce the response to right or wrong. The reviewed contract checks these states independently.
Solve f′(x) = 0
Set the derivative to zero and solve for every x-value.
Coordinates and classification
Substitute back for each y, then evaluate f″(x) at each point to classify it.
Why it goes wrong
The diagnosis stays uncertainty-safe.
A wrong response can support a specific measured diagnosis. A blind multiple-choice diagnostic records only a possible cause until later working provides stronger evidence.
01
The error
Stopping at the x-values, or classifying by assuming a cubic goes maximum then minimum instead of evaluating f″(x) at each point.
02
Why it survives revision
For a cubic with a positive leading coefficient the assumption is correct, and that is the form used in nearly every worked example. The shortcut agrees with the test every time it is practised, and inverts the moment the coefficient is negative.
The worked repair
Repair the smallest broken step.
Teaching is not evidence. This one move is the repair; the proof checks begin after it disappears.
Substitute each x back to get its y, then evaluate f″(x) at each point and read the sign.
What happens after the repair
Correction is the start, not the result.
Immediate
Asked again straight away, on a different question
The repair leaves the screen and a new authored item checks the method unassisted. Being taught something is never evidence that it was learned, so this is the first stage that counts at all.
Delayed
Asked again after at least 1 day
A parallel form of the same method, unassisted, with the repair no longer on screen. A retest in the same session does not count.
Transfer
Asked again when the surface changes
The transfer form uses a negative leading coefficient, or includes a point where f″(x) = 0 so the test is inconclusive and has to be replaced.
Exam
Asked again under exam conditions
Timed, carrying the mark tariff, in an exam-like response format and aligned to the specification. Any one of those missing and the evidence is refused.
REVIEWED BOUNDARY
This page covers one reviewed family: 4.3/4.6/4.7 · Stationary points. Differentiate a cubic, solve the stationary condition, recover both coordinates, and classify each point with the second derivative. Other families within the same GCSE Further Maths topic are outside this reviewed journey, and Studara does not apply the proof label to material that has not been reviewed. No grade-uplift claim is made anywhere on this site.
Straight answers
Before you use this guide.
How do I classify a stationary point?
Evaluate the second derivative there. Positive means a minimum, negative means a maximum. If it is zero the test is inconclusive and you need another method.
Is a cubic always maximum then minimum?
No. That holds for a positive leading coefficient. With a negative leading coefficient the order reverses, which is why the second-derivative test is the reliable route.
Does Studara guarantee a higher grade?
No. Studara has not run a controlled outcome study and publishes no grade-uplift claim. What has been measured, what is withheld, and the metric that failed its bar are all on the public evidence page.
Turn this method into evidence