Question

The minimum value of the function \( x^2 + y^2 + z^2 \) if \( x + y + z = 3a \)

• A. \( 3a \)
• B. \( 3a^2 \)
• C. \( 3a^3 \)
• D. \( 3a^4 \)

Hint:

Use Lagrange multipliers to minimize or maximize multivariable functions with constraints.

Answer 1

The Correct Option is B

We minimize \( x^2 + y^2 + z^2 \) subject to the constraint \( x + y + z = 3a \) using the method of Lagrange multipliers. Let: \[ f(x, y, z) = x^2 + y^2 + z^2,\quad g(x, y, z) = x + y + z - 3a \] Using the method: \[ \nabla f = \lambda \nabla g \Rightarrow 2x = \lambda,\ 2y = \lambda,\ 2z = \lambda \Rightarrow x = y = z \] Substitute in the constraint: \[ x + y + z = 3x = 3a \Rightarrow x = a \Rightarrow f = 3a^2 \]