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HSC Inequalities: AM-GM, Cauchy-Schwarz, and Complex Bounds

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    Vu Hung
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Dominate Hard Proofs with the HSC Inequalities Booklet

Inequalities are often the most daunting part of the NSW HSC Mathematics Extension 2 (and VCE Specialist) syllabus, frequently appearing in the hardest exam sections. The HSC Inequalities booklet is a comprehensive 75-page free resource featuring 55 meticulously worked problems.

Designed for students who want to master algebraic and geometric bounding techniques, this booklet will train your pattern recognition and algebraic manipulation to elite ATAR standards.

What's Inside the HSC Inequalities Booklet?

This booklet covers every major inequality technique required for high-school extension mathematics:

  • Arithmetic-Geometric Mean (AM-GM): Learn to bound sums and products, and cascade AM-GM for multiple variables.
  • Cauchy-Schwarz Inequality: Apply both the discrete sum and integral forms of this powerful theorem.
  • Triangle Inequality: Master vector bounds and complex modulus constraints (∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \le |z_1| + |z_2|).
  • Integration-Based Bounds: Use calculus and areas under curves to prove strict inequalities.
  • Induction Proofs: Prove chains of inequalities and exponential bounds (like Bernoulli's Inequality) using mathematical induction.

How to Maximise Your Study

Inequality questions test your ability to recognize which tool to use. Follow these guidelines:

  1. Check Your Conditions: Never apply AM-GM without explicitly stating that the variables are non-negative.
  2. Equality Conditions: If a question asks "when does equality hold?", don't forget to solve for it (e.g., a=ba=b for AM-GM).
  3. Pick the Right Vectors: When using Cauchy-Schwarz, the hardest step is choosing the correct terms to pair up. Study the worked examples closely.
  4. Draw the Geometry: For the triangle inequality in complex numbers, a quick sketch of the Argand diagram can prevent direction and sign errors.

Practise these techniques separately first, then mix them up to simulate exam conditions. Mastering this booklet will give you a massive edge in the final questions of your HSC or VCE exams!