LEARN · EXPLAIN · REVISE

Minimize the expression over positive x.

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

TOPIC 01

Basic inequality: AM–GM

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

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

01 / Foundation#Your turn

2x+\frac{18}x

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

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.

Teacher preparation and assessment

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

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.