Question

Find the minimum value of x² + y² + z², when x + y + z = 3a.

Answer 1

Let F = (x² + y² + z²) + λ (x + y + z)

We form the equations Fx = 0, F y = 0, Fz = 0

i.e., 2x + λ = 0, 2y + λ = 0, 2z + λ = 0

or λ = – 2x, λ = – 2y, λ = – 2z

⇒ – 2x = – 2y = – 2z or x = y = z

But x + y + z = 3a

Substituting y = z = x, we get 3x = 3a

x = a

∴ x = a, y = a, z = a

The required minimum value of x² + y² + z² is a² + a² + a² = 3a².