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At zero, x³ has which behavior?

Read the idea, work independently, then explain what changed.

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高二選擇性必修第二册(A版).pdf · 5.3 · PDF 89 / printed page 84

Revisit first: Derivative rules and domains

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Predict	Predict what changes in x³−kx as k crosses zero.
8	Build the idea	Sample response: k>0 gives two sign-changing roots; k=0 is a stationary non-extremum.
10	Guided comparison	Sample response: for [−2,1], compare −2,2,−2 and choose maximum 2.
12	Independent practice and discussion	Students contrast endpoint, corner and stationary cases before solving the two transfer tasks.
5	Exit and next step	Exit: name all candidates and explain why a supremum on an open interval need not be a maximum.

Question sequence

Predict what changes in x³−kx as k crosses zero.
Sample response: for [−2,1], compare −2,2,−2 and choose maximum 2.

Board development

Candidates need classification
$f'<0\to f'>0:\ \text{local minimum}$

Expected correct and incorrect responses

Correct sample: Stationary but no extremum
Anticipated wrong response: Local minimum — Calls every stationary point a minimum without checking signs.
Anticipated wrong response: 0 — Checks only stationary points and omits interval endpoints.
Calling every stationary point an extremum.
Reporting a limit at an excluded endpoint as an attained maximum.

Follow-up questions

Follow-up: Compare derivative signs on both sides. Then explain why Local minimum fails.
Follow-up: Compare both endpoints and the stationary point. Then explain why 0 fails.
Change the visible interval and explain why a global optimum also needs endpoint comparisons.

Remediation

Sample response: k>0 gives two sign-changing roots; k=0 is a stationary non-extremum.
Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.

Exit check

Exit: name all candidates and explain why a supremum on an open interval need not be a maximum.

Assessment checklist and sample responses

Representation and conditions
On a closed interval I must compare included endpoints with stationary points inside it.
Connected mathematical reasoning
For x³−3x the derivative signs are +,−,+, so the local maximum is (−1,2).
Interpretation and boundary check
On (0,1), 2x−x²<1 but tends to 1: supremum 1, no attained maximum.