Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Find the minimum for x>0.

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

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TOPIC 01

Quadratic functions, equations and inequalities: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

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

01 / Standard#Your turn

Find the minimum for x>0.

x+81xx+\frac{81}x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check nonnegativity, apply AM–GM and verify attainable equality.
Hint 2
The two positive terms have constant product.
Worked solution

  1. Check nonnegativity, apply AM–GM and verify attainable equality.

  2. Calculate or simplify this relation.

    x+81x≥281=18;x=9x+\frac{81}x\ge2\sqrt{81}=18;\quad x=9

  3. Equal terms give an admissible equality case.

The requested value is 18.

Checks and common pitfalls: Equal terms give an admissible equality case.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗