Overview

Cauchy-Schwarz provides a clean upper or lower bound for sums of products. It is a staple in inequality and geometry problems.

Key Ideas

• $(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2) \ge (a_1b_1+\cdots+a_nb_n)^2$.
• Equality holds when the vectors are proportional.
• Use it to bound dot products or sum-of-squares expressions.

Core Skills

Match Terms Carefully

Decide which expressions should be the $a_i$ and $b_i$ lists so that the right side becomes what you want to bound.

Use the Engel Form

For sums like $\sum \frac{x_i^2}{a_i}$, use $\left(\sum \frac{x_i^2}{a_i}\right) \left(\sum a_i\right) \ge (\sum x_i)^2$.

Check Equality

Equality happens when $a_i/b_i$ is constant. Use it to identify where an extremum is achieved.

Worked Example

Show that $a^2+b^2 \ge \frac{(a+b)^2}{2}$.

Apply Cauchy-Schwarz to $(a,b)$ and $(1,1)$: $(a^2+b^2)(1^2+1^2) \ge (a+b)^2$. So $2(a^2+b^2) \ge (a+b)^2$, or $a^2+b^2 \ge \frac{(a+b)^2}{2}$.

More Examples

Example 1: Two-Term Bound

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

This is the same inequality with $a=x,b=y$.

Example 2: Engel Form

Prove $\frac{1}{a} + \frac{1}{b} \ge \frac{4}{a+b}$ for $a,b>0$.

Use $\left(\frac{1}{a}+\frac{1}{b}\right)(a+b) \ge (1+1)^2 = 4$.

Example 3: Dot Product Bound

If $a_1^2+a_2^2=13$ and $b_1^2+b_2^2=5$, find the maximum of $a_1b_1+a_2b_2$.

By Cauchy-Schwarz, it is at most $\sqrt{13\cdot 5} = \sqrt{65}$.

Strategy Checklist

• Decide which terms to pair as $a_i$ and $b_i$.
• Consider the Engel form for sums of fractions.
• Use equality conditions to confirm the extremum.

Common Pitfalls

• Using Cauchy-Schwarz when AM-GM or Jensen is simpler.
• Missing the equality condition in optimization problems.
• Choosing a pairing that complicates the right-hand side.

Practice Problems