The endpoint nobody checks, and why the switchover on this curve is exact.
Ask for the greatest value while b is at 3, then drag. With the interval ending at 3 the answer is the local maximum, 16, and the class is right. Keep dragging. At b = 4 the endpoint draws level, and past it the greatest value is somewhere the gradient is not zero at all.
The red ring jumps from the turning point to the end of the interval in front of them. That is the moment: differentiating alone can never find an endpoint, because nothing happens to the gradient there.
f(x) = x³ − 12x has its local maximum at (−2, 16), and f(4) = 64 − 48 = 16 as well. So the switchover is exact and lands on a whole number. On a curve with a messy crossover the moment is a blur and the point is lost.
| Stationary points | f′(x) = 3x² − 12 = 0 gives x = ±2. Local maximum (−2, 16), local minimum (2, −16). |
| Sign table | f′(−3) = 15, f′(0) = −12, f′(3) = 15. Up, down, up. |
| Endpoints | f(−3) = 9, f(3) = −9, f(4) = 16, f(5) = 65. |
| 1. y of the local minimum | −16. |
| 2. Greatest on [−3, 5] | B, 65, at the right-hand end. |
| 3. "The maximum is x = −2" | B. That is the location; the value is 16 and the point is (−2, 16). |
1 markDifferentiating and solving f′(x) = 0.
1 markSubstituting back into f. Half the class stops at the x values.
1 markClassifying, with a sign check that actually appears on the page.
1 markOn an interval question, evaluating at the endpoints too and comparing the whole list.
| They say | What it means |
|---|---|
| 2 (Q1) | Gave the x coordinate. The habit this page exists to break. |
| 16 (Q1) | Right value, wrong turning point. The work was done; the labelling was not. |
| 0 (Q1) | Substituted into f′ rather than f. Say it plainly: f′ gives gradients, f gives heights. |
| 16 (Q2) | The target. They found the local maximum and stopped. Go back to the slider rather than explaining again. |
| "Nothing wrong" (Q3) | They do not yet distinguish a location from a value. Very common, and it quietly costs marks in every later topic too. |
"Why check the ends? The gradient is not zero there." Exactly the right observation, and the reason endpoints are missed. A maximum on a closed interval does not have to be smooth; it can be a corner of the domain.
"Is it a maximum or a local maximum?" Encourage the word local always. It costs nothing and it is the honest description.
"Can there be no stationary points?" Yes, and it is worth ten seconds: f′(x) = 3x² + 1 is never zero.
| What is happening | |
|---|---|
| 1 | Recap 5.2: the sign of f′ and what zero means. Keep it short; this is the application lesson. |
| 2 | The full method on x³ − 12x, written out with coordinates as pairs. Insist on the substitution line. |
| 3 | The animation. Greatest value at b = 3, then drag past 4. |
| 4 | Interval questions properly: list the stationary points and both ends, then compare. |
| 5 | Questions 1 to 3. Question 3 is about precision of language and is worth discussing aloud. |
| 6 | Set practice where at least one question gives a domain. |
Do not say "find the maximum" when you mean "find the stationary points". The sloppy phrasing is exactly what produces the x-only answer and the missed endpoint.
Do not introduce the second derivative test. It is Higher Level, it is another thing to get wrong, and the sign check is sufficient and clearer about what is actually happening.
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.
The interval starts at −3 and only the right end moves, which keeps the demonstration simple. No external library, nothing stored, nothing sent.