The second derivative test is a classification tool used in IB Mathematics: Analysis and Approaches (AA) at Higher Level to decide whether a stationary point of a differentiable function is a local maximum, a local minimum, or neither. It relies on the sign of the second derivative evaluated at a critical point where the first derivative equals zero. For IB candidates, the test is not merely a recipe to memorise; it is one of the standard justifications examiners expect when a question asks to determine the nature of stationary points, and it sits at the heart of the optimisation and curve-sketching items that recur on Paper 2.
Across the AA HL syllabus, the test connects three strands: differentiation, function behaviour, and the language of justification. A strong response does more than compute f''(x) and state a sign. It links the algebraic sign to a concavity argument, names the conclusion in correct calculus vocabulary, and acknowledges the inconclusive case honestly when f''(c) = 0. This article walks through the underlying theorem, the three conclusive cases, the inconclusive fourth case, the typical IB question formats, and the tactical habits that separate a Band 6 answer from a Band 5 answer.
The theorem behind the test
Let f be a function defined on an interval containing c, and suppose f is twice differentiable at c. If f'(c) = 0, then c is a stationary point. The second derivative test classifies that stationary point by evaluating f''(c):
- If f''(c) > 0, then f is locally concave up at c, and c is a local minimum.
- If f''(c) < 0, then f is locally concave down at c, and c is a local maximum.
- If f''(c) = 0, the test is inconclusive, and another method is required.
The proof in IB-style questions is rarely required, but the reasoning should be visible. The mean value theorem applied to f' on a small interval around c shows that, when f'(c) = 0 and f'' is continuous, the sign of f'' controls whether f' is increasing or decreasing through zero. If f'' is positive near c, then f' rises from negative to positive as x passes through c, which forces a local minimum. The symmetric argument gives a local maximum when f'' is negative.
IB examiners reward candidates who connect the algebraic sign to a verbal argument. A common marker comment on a borderline Band 5 script is that the candidate wrote f''(c) > 0, therefore minimum, with no mention of concavity. A small phrase such as since f''(c) > 0, the graph is concave up at x = c, so the stationary point is a local minimum lifts the response. Concavity language is part of the rubric's communication criterion and is consistently credited on Paper 2.
It is worth noting that the test assumes f'' exists and is continuous at c. Most functions encountered at AA HL, including polynomials, rational functions with non-vanishing denominators, exponentials, and trigonometric forms on open intervals, satisfy this assumption. The hidden failure case appears in piecewise or restricted-domain functions, which is why IB items occasionally insert a piecewise twist to see whether the candidate checks the assumption before applying the test.
Worked example: cubic with a clean maximum
Consider f(x) = x³ − 3x² + 2. The first derivative f'(x) = 3x² − 6x = 3x(x − 2) vanishes at x = 0 and x = 2. Applying the second derivative test:
- Compute f''(x) = 6x − 6.
- Evaluate at x = 0: f''(0) = −6, which is negative, so x = 0 is a local maximum. The function value is f(0) = 2.
- Evaluate at x = 2: f''(2) = 6, which is positive, so x = 2 is a local minimum. The function value is f(2) = −2.
This is the cleanest case IB Paper 2 will present: a polynomial whose critical points are obvious factors of f'(x). The marks for classification are usually one or two, and the marks for the function values are separate. A typical question might be worth four to six marks: one for finding f'(x), one for solving f'(x) = 0, one each for applying the test at the two points, and one each for the y-coordinates. Candidates who skip the y-coordinates lose a mark even when the classification is correct.
From a tactical angle, the order of operations matters. Solve the critical-point equation, then compute f'', then evaluate. Writing f'' first and trying to back-solve is a recipe for sign errors. Candidates who misread 6x − 6 at x = 0 as 6 rather than −6 are surprisingly common; the remedy is to substitute before deciding the sign.
The inconclusive case and how IB questions disguise it
When f''(c) = 0, the test yields no information. A common IB trap is f(x) = x⁴ at x = 0, where f'(0) = 0 and f''(0) = 0. The test cannot tell us whether the origin is a minimum, a maximum, or an inflection. The correct response is to invoke the first derivative test or to inspect the sign of f' on each side of c, or to factor the function and read the shape directly.
Three disguises appear in exam-style items. The first is the quartic with a repeated root, such as f(x) = (x − 1)⁴, where the second derivative also vanishes. The second is the exponential-cosine product f(x) = eˣ cos x at points where both f' and f'' vanish simultaneously because of trigonometric coincidence. The third is a piecewise function that meets a smoothness condition just barely: the function is differentiable at the join, but the second derivative is undefined or has a jump. In each case, the candidate who mechanically applies the test will write inconclusive, which is a defensible answer, but will not earn the follow-up mark unless they classify the point using an alternative method.
For the first derivative test on AA HL Paper 2, the standard table looks like this:
| Interval around c | Sign of f'(x) | Behaviour of f | Classification |
|---|---|---|---|
| Just left of c | Negative | Decreasing | Local minimum |
| Just right of c | Positive | Increasing | |
| Just left of c | Positive | Increasing | Local maximum |
| Just right of c | Negative | Decreasing | |
| Same sign on both sides | Either + or − | Monotone locally | No extremum, possible inflection |
Writing this table in a solution, even briefly, is a reliable way to convert the inconclusive case into a full-mark response. Examiners respond well to a sign chart because it shows the candidate has thought about the function's behaviour globally, not just at a single point.
Common pitfalls and how to avoid them
Across hundreds of marked scripts, the same five mistakes appear in second-derivative-test items. Each one costs at least one mark, and the first two typically cost two or more.
- Confusing the sign of f'' with the sign of f'. Candidates occasionally write f''(c) > 0, so f is increasing, so it is a maximum. The fix is to remember that f'' describes concavity, not monotonicity. A short mnemonic that works for me: concave up = cup, hold water, minimum at the bottom of the cup.
- Forgetting to find the y-coordinate. Paper 2 marking schemes almost always allocate a mark for the function value at the stationary point. The pair (x, f(x)) is the answer, not x alone.
- Applying the test when the function is not twice differentiable. A piecewise definition can leave f''(c) undefined even when f'(c) = 0. The candidate should glance at the definition before reaching for f''.
- Reporting inconclusive as a final answer without follow-up. A bare inconclusive earns partial credit but rarely full credit. Add a first-derivative sign chart or factor the function locally to finish the classification.
- Arithmetic slips in the second derivative. Differentiating products, quotients, and chains a second time doubles the surface area for error. Re-differentiate from scratch on scrap paper rather than trusting memory of f'.
Each of these is preventable with a thirty-second habit. Before writing the conclusion, re-read the function, recompute f'' at the point, and check that the sign matches the verb. This single discipline accounts for the difference between a 5 and a 6 in many borderline scripts.
IB question types that centre on the test
Three recurring question shapes use the second derivative test as the pivot. Recognising the shape is half the battle, because it tells the candidate which supporting work is required for full marks.
Shape A: classify and sketch. A polynomial or simple rational function is given; the candidate is asked to find stationary points, classify them, and sketch the curve. The test appears as a two- or three-mark sub-step inside a six- or seven-mark item. The classifier marks are awarded for stating the test, applying it correctly, and naming the point. The sketch marks require the candidate to draw a curve that is consistent with the classification, including correct concavity on each side.
Shape B: optimisation with a domain constraint. A real-world scenario is reduced to an expression, and the candidate must find the optimal value. The second derivative test is used to confirm that the critical point inside the domain is a true maximum or minimum rather than just a stationary point. Examiners often add a one-mark sub-question: show that your answer is a maximum. This is the cue to deploy the second derivative test, even if the question does not name it.
Shape C: reasoning about a function given its derivatives. The question states f'(x) and f''(x) without giving f itself, or it gives a graph of f'' and asks the candidate to deduce properties of f. The test still applies, but the algebra is replaced by sign reading. A typical item shows a graph of f'' crossing the x-axis at x = a, x = b, x = c, with f'(a) = 0, f'(b) = 0, f'(c) unknown. The candidate must decide which stationary point is a maximum, which is a minimum, and whether the third is a stationary point at all.