Find the minimum and maximum of f(x, y,z)=x^2+y^2+z^2 subject to two constraints, x+2y+z=4 and x-y=8.

The Lagrangian for this function and the given constraints is



which has partial derivatives (set equal to 0) satisfying



This is a fairly standard linear system. Solving yields Lagrange multipliers of and , and at the same time we find only one critical point at .

Check the Hessian for , given by



is positive definite, since for any vector , which means attains a minimum value of at . There is no maximum over the given constraints.

The answer is b: step 2.