For positive x, y, z with x + y + z = 12, find the minimum of 1/x + 4/y + 9/z and the values of x, y, z that attain it.

For positive x, y, z with x + y + z = 12, find the minimum of 1/x + 4/y + 9/z and the values of x, y, z that attain it.

Approach: Multiply the quantity to be minimised by the constant sum and apply Cauchy-Schwarz in the Engel form, then read the equality condition off the proportionality it requires.

3. The Cauchy-Schwarz inequality in the form (a_1^2/x + a_2^2/y + a_3^2/z)(x + y + z) >= (a_1 + a_2 + a_3)^2 with a = (1, 2, 3) gives (1/x + 4/y + 9/z) * 12 >= (1 + 2 + 3)^2 = 36, so the constrained minimum is 36/12 = 3. The equality condition of Cauchy-Schwarz is proportionality of the two vectors, here a_i/sqrt(x_i) proportional to sqrt(x_i), which means x : y : z = 1 : 2 : 3. With the sum fixed at 12 that gives x = 2, y = 4, z = 6, and checking, 1/2 + 4/4 + 9/6 = 0.5 + 1 + 1.5 = 3. The same inequality is what shows that a Lagrange multiplier solution here is a genuine minimum rather than only a stationary point.

Follow-up: What is the minimum of the sum of a_i^2 / x_i subject to sum x_i = S for general positive a_i, and how does the answer relate to inverse variance weighting?

Key concepts: cauchy-schwarz inequality, equality condition, constrained minimum, proportionality.