⚖️ Inequalities AM-GM & Cauchy

Master inequality techniques using Arithmetic-Geometric Mean and Cauchy-Schwarz inequalities.

🎯 Recognition Cues

Look for these patterns:

  • Optimization: “find minimum/maximum of…”
  • Symmetric expressions: $x + y + z$ with $xyz = k$
  • Sums of squares: $x^2 + y^2 + z^2$ with constraints
  • Fractional expressions: $\frac{a}{b} + \frac{b}{a}$
  • Geometric applications: areas, volumes, distances

📋 Template Solution (4 Steps)

  1. Identify the inequality type (AM-GM vs Cauchy-Schwarz)
  2. Set up the inequality with proper grouping
  3. Apply the inequality theorem
  4. Check equality conditions

🔍 Common Patterns

Pattern 1: Basic AM-GM

Template: $\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$

Example: Find minimum of $x + \frac{1}{x}$ for $x > 0$

Solution:

  1. Apply AM-GM: $\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1$
  2. Solve: $x + \frac{1}{x} \geq 2$
  3. Equality: When $x = \frac{1}{x}$ → $x = 1$
  4. Answer: Minimum is 2, achieved when $x = 1$

Pattern 2: Weighted AM-GM

Template: Use different weights for different terms

Example: If $a, b, c > 0$ and $a + b + c = 1$, find minimum of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Solution:

  1. Apply AM-GM: $\frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{3} \geq \sqrt[3]{\frac{1}{abc}}$
  2. Also: $\frac{a + b + c}{3} \geq \sqrt[3]{abc}$ → $abc \leq \frac{1}{27}$
  3. Combine: $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 3\sqrt[3]{27} = 9$
  4. Equality: When $a = b = c = \frac{1}{3}$

Pattern 3: Cauchy-Schwarz

Template: $(\sum a_i b_i)^2 \leq (\sum a_i^2)(\sum b_i^2)$

Example: Find maximum of $2x + 3y$ subject to $x^2 + y^2 = 1$

Solution:

  1. Apply Cauchy-Schwarz: $(2x + 3y)^2 \leq (2^2 + 3^2)(x^2 + y^2) = 13 \cdot 1 = 13$
  2. Solve: $2x + 3y \leq \sqrt{13}$
  3. Equality: When $\frac{2}{x} = \frac{3}{y}$ and $x^2 + y^2 = 1$
  4. Answer: Maximum is $\sqrt{13}$

Pattern 4: Constrained Optimization

Template: Use Lagrange multipliers or substitution

Example: Find maximum of $xy$ subject to $x + y = 6$

Solution:

  1. Substitute: $y = 6 - x$
  2. Function: $f(x) = x(6-x) = 6x - x^2$
  3. Maximum: $f’(x) = 6 - 2x = 0$ → $x = 3$ → $y = 3$
  4. Answer: Maximum is $3 \cdot 3 = 9$

🎯 AMC-Style Worked Example

Problem: Find the minimum value of $\frac{x^2 + y^2}{xy}$ for positive real numbers $x, y$.

Solution:

  1. Simplify: $\frac{x^2 + y^2}{xy} = \frac{x}{y} + \frac{y}{x}$

  2. Apply AM-GM: $\frac{\frac{x}{y} + \frac{y}{x}}{2} \geq \sqrt{\frac{x}{y} \cdot \frac{y}{x}} = 1$

  3. Solve: $\frac{x}{y} + \frac{y}{x} \geq 2$

  4. Equality condition: When $\frac{x}{y} = \frac{y}{x}$ → $x = y$

  5. Verify: When $x = y$, we get $\frac{x^2 + x^2}{x^2} = 2$ ✓

Answer: Minimum value is 2, achieved when $x = y$

📊 Advanced Techniques

Technique 1: Homogenization

Make expressions homogeneous by introducing appropriate factors.

Technique 2: Substitution

Use clever substitutions to simplify expressions.

Technique 3: Symmetric Sums

Use symmetric sum identities for complex expressions.

Example: Find minimum of $a^2 + b^2 + c^2$ subject to $a + b + c = 1$

Solution:

  1. Use identity: $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+bc+ca)$
  2. Substitute: $a^2 + b^2 + c^2 = 1 - 2(ab+bc+ca)$
  3. Minimize: Need to maximize $ab+bc+ca$
  4. AM-GM: $ab+bc+ca \leq \frac{(a+b+c)^2}{3} = \frac{1}{3}$
  5. Answer: Minimum is $1 - 2 \cdot \frac{1}{3} = \frac{1}{3}$

⚠️ Common Pitfalls

Pitfall: Wrong equality conditions

Fix: Always check when equality holds in your inequality.

Pitfall: Forgetting constraints

Fix: Make sure your solution satisfies all given constraints.

Pitfall: Wrong application of Cauchy-Schwarz

Fix: Ensure you’re applying it to the right vectors.

Pitfall: Not checking if minimum/maximum exists

Fix: Verify that your critical point actually gives the desired extremum.

📋 Quick Reference

AM-GM Inequality

For positive real numbers $a_1, a_2, \ldots, a_n$: $$\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$$ Equality when all numbers are equal.

Cauchy-Schwarz Inequality

$$(\sum_{i=1}^n a_i b_i)^2 \leq (\sum_{i=1}^n a_i^2)(\sum_{i=1}^n b_i^2)$$ Equality when vectors are proportional.

Common Applications

  • Minimum of sum: Use AM-GM on individual terms
  • Maximum of dot product: Use Cauchy-Schwarz
  • Constrained optimization: Combine with substitution

Equality Conditions

  • AM-GM: All terms equal
  • Cauchy-Schwarz: Vectors proportional
  • Weighted AM-GM: Terms proportional to weights

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