Cue Column:

• When to apply AM-GM?
• What is the equality condition?
• How does it help in finding min/max?

Note Column: AM-GM inequality: For positive reals $a_1$, $a_2$, ..., $a_n$: $\frac{a_1+a_2+...+a_n}{n}$ >= ($a_1$$a_2$...*$a_n$)^{1/n}

Equality iff $a_1$ = $a_2$ = ... = $a_n$.

Finding minimum of sum given constant product: If ab = k (constant), min(a+b) = 2*sqrt(k), achieved when a = b = sqrt(k).

Finding minimum of a + k/a for a > 0: By AM-GM: a + k/a >= 2*sqrt(k), equality when a = sqrt(k).

JEE application: To find the minimum value of ($x^2$ + 1/$x^2$) for x > 0: $x^2$ + 1/$x^2$ >= 2*sqrt(1) = 2, equality when $x^2$ = 1, i.e., x = 1.

Extended: For weighted AM-GM, and for multiple variable optimization, the technique extends naturally.

Summary: AM-GM gives minimum of sum (given product is fixed) and maximum of product (given sum is fixed). Always check equality condition.