Overview

AM-GM is the fastest way to bound expressions with fixed products or sums. It also provides clean equality conditions.

Key Ideas

• For nonnegative $a,b$, $\frac{a+b}{2} \ge \sqrt{ab}$.

  • In general, for nonnegative $a_1, a_2, ... a_n$, $\frac{a_1 + a_2 + ... + a_n}{n} \ge \sqrt[n]{a_1 a_2 ... a_n}$
• Equality holds when all variables are equal.

  • i.e. When $a_1 = a_2 = ... = a_n$, $\frac{a_1 + a_2 + ... + a_n}{n} = \sqrt[n]{a_1 a_2 ... a_n}$
• For $ax+\frac{b}{x}$, the minimum is $2\sqrt{ab}$ at $x=\sqrt{\frac{b}{a}}$.

Core Skills

Check Nonnegativity

AM-GM requires nonnegative terms. If variables can be negative, reframe the expression (for example with squares) before applying it.

Normalize Expressions

Rewrite the expression to look like a sum of terms with a fixed product. Then apply AM-GM directly.

Use Equality Cases

The equality condition tells you where the minimum or maximum occurs. Use it to solve for parameters.

Worked Example

Find the minimum of $x + \frac{4}{x}$ for $x>0$.

By AM-GM: $\frac{x + \frac{4}{x}}{2} \ge \sqrt{x \cdot \frac{4}{x}} = 2$ So $x + \frac{4}{x} \ge 4$, with equality at $x=2$.

More Examples

Example 1: Three Variables

For $a,b,c \ge 0$ with $abc=1$, find the minimum of $a+b+c$.

By AM-GM, $a+b+c \ge 3\sqrt[3]{abc} = 3$.

This is when $a = b = c = 1$.

Example 2: Fixed Sum

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

By AM-GM, $\frac{x+y}{2} \ge \sqrt{xy}$, so $xy \le 25$ at $x=y=5$.

Example 3: Weighted Form

Find the minimum of $2x + \frac{8}{x}$ for $x>0$.

By AM-GM, $2x + \frac{8}{x} \ge 2 \sqrt{2x \cdot \frac{8}{x}} = \sqrt{16} = 4$

This is when $2x = \frac{8}{x}$, so when $x = 2$ as $x>0$

Example 4: Extensive Use

Find the minimum value of $x + y$ when $(x - 2)(y - 2) = 81$ for $x, y > 2$

From the given constraints, $x, y > 2$, rewrite this into $x - 2, y - 2, > 0$ and then apply AM-GM:

$(x-2) + (y-2) \ge 2 \sqrt{(x-2)(y-2)} = 2 \sqrt{81} = 18$

Which becomes:

$x + y \ge 22$

The equality is when $x - 2 = y - 2$ which is when, $x = y = 11$.

Strategy Checklist

• Confirm all terms are nonnegative.
• Rewrite to expose a fixed product or fixed sum.
• Apply AM-GM and note when equality holds.
• Verify the equality point satisfies the domain.

Common Pitfalls

• Applying AM-GM to negative terms.
• Forgetting to check the equality condition.
• Mixing up minimum and maximum in fixed-sum/product problems.

Practice Problems