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Inequalities

AM-GM, Cauchy-Schwarz, triangle inequality, and optimisation problems

Key Concepts
  • AM–GM: the arithmetic mean is at least the geometric mean
  • Cauchy–Schwarz bounds sums of products
  • The triangle inequality and its variants
  • Equality conditions are often the key to the problem
Important Formulae
AM–GM (a+b)/2 ≥ √(ab), equality when a = b
Cauchy–Schwarz (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)
Power mean / QM–AM √((a²+b²)/2) ≥ (a+b)/2
Quick Tips
  • Equality in AM–GM holds only when all the terms are equal.
  • Normalising (e.g. setting a+b+c = 1) often simplifies an inequality.
Sample Practice Questions

  1. Solve x² < 4x.

    • x < 0 or x > 4
    • 0 < x < 4
    • x > 4
    • -4 < x < 0
    Show answer

    Answer: 0 < x < 4

  2. Solve: x/(x-2) > 3

    • x > 3
    • x < 2
    • 2 < x < 3
    • x < 2 or x > 6
    Show answer

    Answer: 2 < x < 3

  3. Cauchy-Schwarz inequality states (Σaᵢbᵢ)² ≤ ?

    • (Σaᵢ)²(Σbᵢ)²
    • (Σaᵢ²)(Σbᵢ²)
    • Σaᵢ²bᵢ²
    • (Σaᵢbᵢ²)
    Show answer

    Answer: (Σaᵢ²)(Σbᵢ²)

  4. For which values of k is kx² + 2x + k > 0 for all x?

    • k > 0
    • k > 1
    • k ≥ 1
    • k > 2
    Show answer

    Answer: k > 1

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Practice Questions

Practise PRMO questions on Inequalities. Answers are revealed after each question.

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Algebra