g10-05-08 · Core topics
Can the largest value be reached?
Does y=x on (0,2) have a maximum just because the graph approaches 2?
1 MIN · PREDICT ON PAPER
Write a candidate maximum; predict what happens when the endpoint is closed.
Discuss or write first. No typed answer is required to continue.
Domain [0, 6]. Cursor input 2: output 5.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
On (0,2), every allowed x has a larger allowed input (x+2)/2. Thus 2 is a bound but no maximum is attained. On (0,2], the maximum is 2.
Conditions: Extrema depend on the complete stated domain; local peaks or a finite graph window do not establish global extrema.
Mathematical justification: For 0<x<2, x<(x+2)/2<2. This defeats every proposed maximum, not merely the sampled candidates.
1 MIN · EXIT CHECK
State maximum and minimum of y=x on (0,2].
Check after answering
Maximum 2; no minimum, because any allowed x has smaller allowed x/2.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Ask a student to propose a largest value and another to construct a strictly larger allowed value.
Watch for: A curve reaching the top of the visible window has a maximum.
Challenge: Trim the water graph to [3,6]. Explain why its earlier peak no longer counts.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
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