IB Maths AI HL
Further Differentiation
Paper 1 & 2
~8 min read
Concavity & Points of Inflection
The second derivative does more than classify turning points — it describes the shape of the whole curve. Where f′′(x) > 0 the curve bends like a smile (concave up); where f′′(x) < 0 it bends like a frown (concave down). A point of inflection is where that bend flips — and, crucially, it doesn’t have to be a stationary point.
📘 What you need to know
- Concave up: f′′(x) > 0 on an interval — curves like a smile ⌣.
- Concave down: f′′(x) < 0 on an interval — curves like a frown ⌢.
- Point of inflection: where the curve changes concavity.
- Both conditions needed: f′′(x) = 0 AND the concavity actually changes (the sign of f′′ flips).
- Not necessarily stationary: a point of inflection doesn’t require f′(x) = 0.
- Horizontal inflection: if it is stationary too, the tangent is horizontal.
Concavity
Concavity is the way a curve bends. The second derivative reads it off directly — its sign over an interval tells you which way the curve cups.
concave up
⌣ smile
f′′(x) > 0 — tangents lie below the curve.
concave down
⌢ frown
f′′(x) < 0 — tangents lie above the curve.
Concavity from the second derivative
f′′(x) > 0 → concave up · f′′(x) < 0 → concave down
✗ not in booklet — learn the smile/frown link
🧠 “Up is a happy smile, down is a sad frown”
Concave up ⌣ is the mouth of a happy face (f′′ > 0). Concave down ⌢ is the mouth of a sad face (f′′ < 0). The sign of the second derivative is the shape of the mouth.
Points of inflection
A point of inflection is where the curve switches from one concavity to the other. It needs both conditions to hold — a zero second derivative alone is not enough.
Conditions for a point of inflection
f′′(x) = 0 AND concavity changes (sign of f′′ flips)
✗ both conditions required — the second is essential
🤔 Why isn’t f′′(x) = 0 enough by itself?
A point where f′′ = 0 could be an inflection — but it could equally be a local max or min where the curvature momentarily flattens (think of y = x4 at the origin: f′′(0) = 0, yet it’s a minimum). What makes it an inflection is that the concavity actually changes — f′′ must switch sign through the point. The other direction is reliable, though: if you already know there’s an inflection at x = a, then f′′(a) = 0 must hold.
🧭 Recipe — finding a point of inflection
- Differentiate twice to get f′′(x), and solve f′′(x) = 0 for candidate x-values.
- Test concavity either side: check the sign of f′′ just below and just above. If it changes, it’s a point of inflection.
- Find y if required, by substituting x into f(x).
Non-stationary inflections exist: unlike a turning point, an inflection doesn’t need f′(x) = 0. The normal distribution curve is a classic example — it has two points of inflection, neither of them stationary.
Worked examples
WE 1For f(x) = x3 − 3x + 2, is the curve concave up or down at x = −2 and x = 2?
Find f′′(x) and read the sign at each point.
f′(x) = 3x² − 3, f″(x) = 6x
f″(−2) = −12 < 0 → concave down
f″(2) = 12 > 0 → concave up
x = −2: concave down; x = 2: concave up
WE 2For which x is that curve concave up?
Solve f′′(x) > 0.
concave up when f″(x) > 0
6x > 0 → x > 0
concave up for x > 0
WE 3Find f′′(x) for y = 2x3 − 18x2 + 24x + 5 and solve f′′(x) = 0
Differentiate twice and find the candidate x.
f′(x) = 6x² − 36x + 24
f″(x) = 12x − 36
12x − 36 = 0
x = 3 (candidate inflection)
WE 4Justify that x = 3 is a point of inflection, and give its coordinates
Test concavity either side, then find y.
f″(2.9) = 12(2.9) − 36 = −1.2 < 0 (concave down)
f″(3.1) = 12(3.1) − 36 = 1.2 > 0 (concave up)
concavity changes through x = 3 ✓
f(3) = 2(27) − 18(9) + 24(3) + 5 = 54 − 162 + 72 + 5 = −31
(3, −31) is a point of inflection
f″(3) = 0 AND concavity changes — both conditions hold.
WE 5Why is x = 0 not a point of inflection for y = x4?
Check both conditions — the second one fails.
f″(x) = 12x², so f″(0) = 0 (first condition met)
f″(−1) = 12 > 0, f″(1) = 12 > 0
concave up on both sides — no sign change
not an inflection (it’s a minimum)
💡 Top tips
- Smile up, frown down — link the sign of f′′ to the curve’s shape.
- Always test the sign change — f′′ = 0 alone never confirms an inflection.
- Inflections needn’t be stationary — don’t expect f′(x) = 0.
- Graph f′′ and find where it crosses (not just touches) the x-axis to locate inflections.
- Give coordinates — substitute back into f(x) for the y-value.
- Use the GDC to sketch and sanity-check.
⚠ Common mistakes
- Calling every f′′ = 0 an inflection — it could be a max or min; check the sign change.
- Confusing up and down — concave up is f′′ > 0 (smile).
- Expecting f′(x) = 0 — most inflections are not stationary.
- Touching vs crossing — on the f′′ graph, only a crossing of the axis marks a concavity change.
- Skipping the justification — “fully justify” wants the sign test, not just f′′ = 0.
That wraps up Further Differentiation. The whole unit grew from one engine — the derivative — extended in every direction. You learned to differentiate the special functions (sin, cos, tan, ex, ln x), then the three combining rules — chain (functions inside functions), product (functions multiplied), and quotient (functions divided). From there you applied differentiation to related rates linked through time, stepped up to the second derivative, and used it to classify stationary points and read concavity and inflections. First derivative for slope, second derivative for shape — hold those two ideas together and the whole of calculus stays one connected picture.
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