Lesson 6: Stationary Points, Concavity & Curve Sketching
What the first and second derivatives tell you about a shape you have not drawn yet: stationary points, their nature, concavity, and where the curve changes its mind.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Stationary points (f'(x) = 0), local maxima, local minima, horizontal points of inflection, second derivative test (f''(x) > 0 for local min, f''(x) < 0 for local max), and concavity changes (f''(x) = 0).
Locate and classify all stationary points using 1st and 2nd derivative tests, determine intervals of concavity, and sketch accurate polynomial/rational curves.
🎥 Media Anchor & Pedagogical Scaffold
Second Derivative Test — Khan Academy
Video (6 min 12 sec, Khan Academy): A visual guide to using the second derivative to classify stationary points as local maxima or minima, and to understand concavity. Clear, direct, and perfect for curve-sketching practice.
🧠 1. Before Viewing (Activate & Predict)
What is the physical and geometric meaning when f'(x) = 0 and f''(x) changes sign?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 6 min 12 sec clip. In your calculus logbook, record:
- The second derivative test: Write: If f''(x₀) > 0, local minimum; if f''(x₀) < 0, local maximum; if f''(x₀) = 0, test is inconclusive.
- A worked example: For f(x) = x³ - 3x, find f'(x) = 3x² - 3, set to 0 to get x = ±1. Then f''(x) = 6x. At x = -1: f''(-1) = -6 < 0 → local max. At x = 1: f''(1) = 6 > 0 → local min.
- Concavity: Note when f'' > 0 (curve is concave up) and when f'' < 0 (concave down).
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Draw concavity memory cues: f''(x) > 0 is 'happy cup' (concave up, local min); f''(x) < 0 is 'frown' (concave down, local max).
Immediate Task: Add to your Level 3 Calculus revision logbook (section 6): Curve Analysis & Stationary Point Classification.
⚡ Whakaoho | Do Now: Solving f′(x) = 0 (5 mins)
Solve 3x² − 12 = 0 and 2x³ − 8x = 0. Then differentiate f(x) = x³ − 3x and differentiate your answer again. Today the second derivative stops being an extra step and starts doing real work.
📖 Activity 1: Full Curve Sketch With Justified Nature (20 mins)
Take f(x) = x³ − 3x. Find the stationary points, then classify each one using f″(x): negative means a maximum, positive a minimum. Find where f″(x) = 0 and check whether concavity actually changes there before calling it a point of inflection. Sketch the curve using only what you derived, then check it against a graphing tool. Where your sketch and the tool disagree, the working tells you which one is wrong.
📝 Activity 2: Exam-Style Practice — Classify and Justify (15 mins)
For f(x) = x⁴ − 4x³: find and classify all stationary points. For Merit, use the second-derivative test and state its result at each point.
For Excellence — the same test, doing real work. Suppose a fishery's stock B (tonnes) grows each year by g(B) = 0.4B(1 − B/8000), and is harvested at a steady h tonnes a year, so the stock holds level wherever g(B) = h. This is a scenario built to be worked by hand — the numbers are illustrative and are not real quota figures.
- In this scenario, find the stationary point of g and use g″ to classify it. (You should get B = 4000 t, exactly half the maximum the fishery can hold, and g″ < 0, so it is a maximum.)
- Show that the largest harvest the scenario can sustain indefinitely is g(4000) = 800 tonnes a year.
- Now suppose h = 900 and try to solve g(B) = 900. You will find no solution. Say plainly, in one sentence, what that means for the fishery — and note that the second-derivative test is what told you the ceiling was there before anyone had to discover it the hard way.
This is what a stationary point is for. A maximum sustainable yield is not a policy opinion; it is g′(B) = 0, and it sits at the point where the rate of removal can just be matched by the rate of natural increase.
🎫 Exit Ticket: Find, Classify, Justify (5 mins)
(a) Find the stationary points of f(x) = x³ − 3x². (b) Classify each one using f″. (c) In one sentence: why is f″(x) = 0 not enough to declare a point of inflection?
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 alignment: Direct preparation for NCEA Level 3 Achievement Standard AS 91578, Apply differentiation methods in solving problems (external, 6 credits), against The New Zealand Curriculum (2007) Mathematics and Statistics Level 8 — Calculus. Emphasise complete algebraic working and a conclusion written in the context of the question, which is what separates Merit from Achieved.
Materials: Grid paper and ruler for the full sketch, graphics calculator, Desmos to check the finished sketch after it is drawn by hand and not before, calculus logbook.
Pacing (56 mins): Do Now 5 · media anchor 11 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5.
Formative assessment — what to look for: Part (c) targets the inflection-versus-turning-point confusion. It shows up directly in curve-sketching questions, where ākonga mark an inflection wherever f″ happens to vanish. If the class is shaky, do the extension example together — one counter-example fixes it faster than a definition.
Differentiation. Entry: Provide the sign table as a printed frame; ākonga fill the signs and read off the nature. On level: Full sketches: intercepts, stationary points with justified nature, concavity, and end behaviour. Extension: Sketch f(x) = x⁴ − 4x³ completely. Then use f(x) = x⁴ to explain why f″(x) = 0 does not by itself guarantee a point of inflection.