If x, y and z are positive real numbers with x + y + z = 1, what is the minimum value of 1/x + 4/y + 9/z?

By the Cauchy–Schwarz inequality, (x + y + z)(1/x + 4/y + 9/z) ≥ (1 + 2 + 3)² = 36.

Since x + y + z = 1, the expression is at least 36. Equality holds when x : y : z = 1 : 2 : 3, i.e. x = 1/6, y = 1/3, z = 1/2, giving 6 + 12 + 18 = 36 ✓.

Why 14 is wrong: 1 + 4 + 9 = 14 would need each fraction's denominator to be 1, impossible when they sum to 1. Why x = y = z = 1/3 fails: that gives 3 + 12 + 27 = 42, above the minimum.

Tip: For minimising Σ aᵢ²/xᵢ subject to Σ xᵢ = S, the minimum is (Σ aᵢ)²/S with xᵢ ∝ aᵢ.