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: (a_1+a_2+...+a_n)/n >= (a_1a_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.