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Maths olympiad lesson: the AM–GM inequality, from rectangles to least values

Name: ____________________ Date: ____________________

Show your working. Check every root by substitution and label your sketches.

AM–GM: practice

Show your working. For every least or greatest value, say where it is reached. The last questions are proofs: write a line for each step, with its reason.

Diagnostic question: choose and explain

  1. x is positive. What is the least value of x + 12/x?
    1. √12
    2. 7
    3. 2√12
    4. 24
    5. There is none: it gets as small as you like

Practice questions

  1. For a = 4 and b = 9: (a + b) ÷ 2 = ___ and √(ab) = ___
  2. x is positive. The least value of x + 49/x is ___, when x = ___
  3. x is positive. The least value of 2x + 50/x is ___, when x = ___
  4. A rectangular pen against a wall has fence on three sides and area 72 m². Its two equal sides are x m. The shortest fence is ___ m, when x = ___
  5. Ravi says: x + 1/x ≥ 2 for every number x except 0. Is he right?
    1. Yes: AM–GM says so
    2. No: x = −1 gives −2
    3. Yes: try x = 2, 3 or 0.5 and it is always more than 2
  6. a and b are positive and (a + b) ÷ 2 = √(ab). What must be true?
    1. a = 0 or b = 0
    2. a + b = 2
    3. a = b
    4. ab = 1
  7. Prove that a/b + b/a ≥ 2 for all positive a and b, and say when the two sides are equal.
  8. Stretch: x and y are positive and xy = 12. What is the least value of 3x + 4y?
  9. Stretch: a, b and c are positive and abc = 1. Prove that (1 + a)(1 + b)(1 + c) ≥ 8. When are the two sides equal?

Support: For the proofs: start from a square that is 0 or more, or from AM–GM for two numbers you choose. Say what you multiply or add, and end with when the two sides are equal.

AM–GM: exit

A few lines: the value, why it can't be less, and where it is reached.

  1. x is positive. Find the least value of 4x + 1/x. Show that it can never be less, and say for which x it is reached.

Exit check

One least value with its proof and its equality case shows who has the whole idea.

x is positive. Find the least value of 4x + 1/x.

AM–GM: least values and proofs

For every least or greatest value, say where it is reached. Write proofs a line at a time, each with its reason.

1. x is positive. Find the least value of x + 64/x, and the x that gives it.

2. A rectangle has area 81 cm². What is the shortest its perimeter can be? Which rectangle gives it?

3. Two positive numbers multiply to 50. What is the least their sum can be? Give it exactly and to two decimal places.

4. Prove that x² + 1/x² ≥ 2 for every x except 0. When are the two sides equal?

5. A rectangular vegetable bed against a wall has an edging on three sides and area 98 m². Its two equal sides are x m. Find the least length of edging, and x.

6. a and b are positive and a + b = 1. Prove that 1/a + 1/b ≥ 4. When are the two sides equal?

7. Challenge: x is positive. Find the least value of x + 4/(x + 1), and the x that gives it. (Hint: the two parts don't multiply to a number. Can you make two that do?)

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