Kenneth ChengMathematical modeling
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Basic inequality: AM–GM

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

高一必修 第一册(A版).pdf · 2.2 · PDF 51 / printed page 44

Revisit first: Properties of equalities and inequalities

TOPIC 01

Basic inequality: AM–GM

Build understanding of basic inequality: am–gm through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Apply basic inequality: am–gm with explicit conditions.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

AM–GM

For nonnegative terms, arithmetic mean is at least geometric mean.

(a+b)/2≥ab(a+b)/2\ge\sqrt{ab}

Equality

The terms must be equal and allowed by the domain.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in basic inequality: am–gm changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

The difference is 2ab=4. Equality holds exactly when ab=0.(a+b)² = 9a²+b² = 5

The difference is 2ab=4. Equality holds exactly when ab=0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the minimum for x>0.

x+4xx+\frac{4}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+4x≥24=4;x=2x+\frac{4}x\ge2\sqrt{4}=4;\quad x=2

  3. Equal terms give an admissible equality case.

The requested value is 4.

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

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

02 / Standard#Worked example

Positive a,b have sum 4; maximize their product.

  • 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
Fix the arithmetic mean.
Worked solution

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

  2. Calculate or simplify this relation.

    ab≤((a+b)/2)2=4;a=b=2ab\le((a+b)/2)^2=4;\quad a=b=2

  3. The equal pair satisfies the required sum.

The requested value is 4.

Checks and common pitfalls: The equal pair satisfies the required sum.

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

03 / Transfer#Worked example

Minimize the sum of two positive numbers with this product.

ab=9ab=9
  • 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
With fixed product, AM–GM gives a lower sum bound.
Worked solution

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

  2. Calculate or simplify this relation.

    a+b≥2ab=6;a=b=3a+b\ge2\sqrt{ab}=6;\quad a=b=3

  3. Give an equality pair to prove attainment.

The requested value is 6.

Checks and common pitfalls: Give an equality pair to prove attainment.

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

04 / Foundation#Your turn

Find the minimum for x>0.

x+9xx+\frac{9}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+9x≥29=6;x=3x+\frac{9}x\ge2\sqrt{9}=6;\quad x=3

  3. Equal terms give an admissible equality case.

The requested value is 6.

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

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

05 / Foundation#Your turn

Positive a,b have sum 6; maximize their product.

  • 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
Fix the arithmetic mean.
Worked solution

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

  2. Calculate or simplify this relation.

    ab≤((a+b)/2)2=9;a=b=3ab\le((a+b)/2)^2=9;\quad a=b=3

  3. The equal pair satisfies the required sum.

The requested value is 9.

Checks and common pitfalls: The equal pair satisfies the required sum.

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

06 / Foundation#Your turn

A rectangle has perimeter 12. Find its largest area.

  • 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
Convert perimeter into a fixed sum of positive sides.
Worked solution

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

  2. Calculate or simplify this relation.

    x+y=6;xy≤9x+y=6;\quad xy\le9

  3. The maximum occurs for a square.

The requested value is 9.

Checks and common pitfalls: The maximum occurs for a square.

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

07 / Foundation#Your turn

Minimize the expression over positive x.

2x+18x2x+\frac{18}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
Include the coefficient two in the product.
Worked solution

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

  2. Calculate or simplify this relation.

    2x+18x≥236=12;x=32x+\frac{18}x\ge2\sqrt{36}=12;\quad x=3

  3. Equality concerns the complete two terms, not just x and 1/x.

The requested value is 12.

Checks and common pitfalls: Equality concerns the complete two terms, not just x and 1/x.

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

08 / Standard#Your turn

Does the function attain a minimum on this domain?

x+9x,x>3x+\frac{9}x,\quad x>3
  • 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 equality point is excluded.
Worked solution

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

  2. Calculate or simplify this relation.

    x+9x−6=(x−3)2x>0x+\frac{9}x-6=\frac{(x-3)^2}{x}>0

  3. Calculate or simplify this relation.

    x→3+⇒x+9x→6x\to3^{+}\Rightarrow x+\frac{9}x\to6

  4. An infimum need not be a minimum.

No; its infimum is 6, approached but not attained.

Checks and common pitfalls: An infimum need not be a minimum.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

Prove AM–GM for nonnegative a,b.

a+b≥2aba+b\ge2\sqrt{ab}
  • 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
Start from a square that cannot be negative.
Worked solution

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

  2. Calculate or simplify this relation.

    (a−b)2=a+b−2ab≥0(\sqrt a-\sqrt b)^2=a+b-2\sqrt{ab}\ge0

  3. Real square roots require the nonnegative assumptions.

Equality holds exactly when a=b.

Checks and common pitfalls: Real square roots require the nonnegative assumptions.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

10 / Standard#Your turn

Minimize the sum of two positive numbers with this product.

ab=16ab=16
  • 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
With fixed product, AM–GM gives a lower sum bound.
Worked solution

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

  2. Calculate or simplify this relation.

    a+b≥2ab=8;a=b=4a+b\ge2\sqrt{ab}=8;\quad a=b=4

  3. Give an equality pair to prove attainment.

The requested value is 8.

Checks and common pitfalls: Give an equality pair to prove attainment.

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

11 / Standard#Your turn

Find the minimum for x>0.

x+16xx+\frac{16}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+16x≥216=8;x=4x+\frac{16}x\ge2\sqrt{16}=8;\quad x=4

  3. Equal terms give an admissible equality case.

The requested value is 8.

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

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

12 / Transfer#Your turn

Positive a,b have sum 8; maximize their product.

  • 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
Fix the arithmetic mean.
Worked solution

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

  2. Calculate or simplify this relation.

    ab≤((a+b)/2)2=16;a=b=4ab\le((a+b)/2)^2=16;\quad a=b=4

  3. The equal pair satisfies the required sum.

The requested value is 16.

Checks and common pitfalls: The equal pair satisfies the required sum.

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

13 / Transfer#Your turn

A rectangle has perimeter 16. Find its largest area.

  • 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
Convert perimeter into a fixed sum of positive sides.
Worked solution

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

  2. Calculate or simplify this relation.

    x+y=8;xy≤16x+y=8;\quad xy\le16

  3. The maximum occurs for a square.

The requested value is 16.

Checks and common pitfalls: The maximum occurs for a square.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for basic inequality: am–gm?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Apply basic inequality: am–gm with explicit conditions.
    • AM–GM: For nonnegative terms, arithmetic mean is at least geometric mean.
      (a+b)/2≥ab(a+b)/2\ge\sqrt{ab}
    • Equality: The terms must be equal and allowed by the domain.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: For nonnegative terms, arithmetic mean is at least geometric mean.
    • Expected reasoning: The terms must be equal and allowed by the domain.
    • Expected correction: A lower bound may be unattainable.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗