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WMA12/01 Oct 2021, Question 9: Proof by Exhaustion and Counter-example

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 4 marks

The question, in short

Show the inequality (x + y)/2 ≥ √(xy) holds for positive x and y, then disprove it for negative values using a counter-example.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Prove that the arithmetic mean of positive x and y is at least the square root of their product.

  1. 1For positive x and y, √x and √y are real, and (√x − √y)² ≥ 0
  2. 2x − 2√(xy) + y ≥ 0
  3. 3x + y ≥ 2√(xy)
  4. 4(x + y)/2 ≥ √(xy)

AnswerShown: (√x − √y)² ≥ 0 ⇒ x − 2√(xy) + y ≥ 0 ⇒ (x + y)/2 ≥ √(xy)

Part (b) · 1 mark

Give a counter-example showing the inequality fails when x and y are both negative.

  1. 1Take x = −8, y = −2
  2. 2(x + y)/2 = −10/2 = −5
  3. 3√(xy) = √16 = 4
  4. 4−5 < 4, so (x + y)/2 ≥ √(xy) does not hold

Answere.g. x = −8, y = −2: (x + y)/2 = −5 but √(xy) = √16 = 4, and −5 < 4, so the inequality fails

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