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Find the maximum of x² on [−1,2].

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

TOPIC 01

Monotonicity and constrained extrema

Use derivative sign changes, endpoints and feasible domains to distinguish local extrema from global optima.

What you will be able to explain

  • Use derivative sign changes, endpoints and feasible domains to distinguish local extrema from global optima.
  • Handle parameter boundaries and non-attained bounds.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Find the maximum of x² on [−1,2].

f=x2; −1≤x≤2f=x^2;\ -1\le x\le2
  • Use the stated real domain and distinguish stationary candidates, attained extrema and actual concavity changes.
Skills and prerequisite lessons

Scope: JM02 · This task uses elementary probability rules or polynomial calculus named in the JAE syllabus.

  1. 0
  2. 1
  3. 2
  4. 4
Working and explanation
BUILD THE REASONING
Hint 1
Compare both endpoints and the stationary point.
Hint 2
Start from this intermediate relation.
f(−1)=1; f(0)=0; f(2)=4f(-1)=1;\ f(0)=0;\ f(2)=4
Worked solution
  1. Compare both endpoints and the stationary point.
  2. Establish and simplify this mathematical relation.
    f(−1)=1; f(0)=0; f(2)=4f(-1)=1;\ f(0)=0;\ f(2)=4
  3. Establish and simplify this mathematical relation.
    max⁡f=4\max f=4
  4. An extremizing coordinate is not the requested value.
4
Checks and common pitfalls: An extremizing coordinate is not the requested value.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5PredictPredict what changes in x³−kx as k crosses zero.
8Build the ideaSample response: k>0 gives two sign-changing roots; k=0 is a stationary non-extremum.
10Guided comparisonSample response: for [−2,1], compare −2,2,−2 and choose maximum 2.
12Independent practice and discussionStudents contrast endpoint, corner and stationary cases before solving the two transfer tasks.
5Exit and next stepExit: 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→f′>0: local minimumf'<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.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
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