Overview

Inequalities are about bounding expressions. Learn to spot when AM-GM or Cauchy applies, and how to rewrite expressions to match known forms.

Key Ideas

• AM-GM: for nonnegative $a,b$, $\frac{a+b}{2}\ge\sqrt{ab}$.
• Cauchy-Schwarz: $(a^2+b^2)(x^2+y^2)\ge (ax+by)^2$.
• Equality cases often guide the right substitution.

Core Skills

Match a Known Inequality

Rewrite the target expression to fit AM-GM, Cauchy-Schwarz, or Jensen. The right form is often the entire battle.

Use Equality Conditions

Solve for when equality holds. This gives the extremal configuration and helps verify the bound.

Normalize Constraints

If $a+b$ or $ab$ is fixed, scale variables or substitute to reduce degrees.

Worked Example

For positive $a,b$, show $a+b \ge 2\sqrt{ab}$.

Apply AM-GM directly. Equality holds when $a=b$.

More Examples

Example 1: Cauchy Bound

Show $(x+y)^2 \le 2(x^2+y^2)$.

Apply Cauchy-Schwarz to $(x,y)$ and $(1,1)$.

Example 2: AM-GM Minimum

Find the minimum of $x+\frac{9}{x}$ for $x>0$.

By AM-GM, the minimum is $2\sqrt{9}=6$ at $x=3$.

Example 3: Boundary Maximum

If $x+y=10$ and $x,y\ge 0$, find the maximum of $x^2+y^2$.

The maximum occurs at an endpoint: $x^2+y^2$ is largest when one is $0$, so the maximum is $100$.

Strategy Checklist

• Identify the relevant inequality and rewrite to match it.
• Check nonnegativity requirements.
• Use equality to confirm the extremum.
• Verify any domain constraints.

Common Pitfalls

• Applying AM-GM to negative terms.
• Missing the equality condition.
• Overlooking that maxima may occur at boundary values.

Practice Problems