Find the minimum and maximum of f(x, y,z)=x2+y2+z2 subject to two constraints, x+2y+z=7 and x−y=6.

Use the method of Lagrange multipliers. We have Lagrangian



with partial derivatives (set equal to 0) of







As , and , we can obtain





From this, we find a single critical point:





At this point, we have a value of



To determine what kind of extremum occurs at this point, we check the Hessian of :



We observe that at any point , and that the eigenvalues of this matrix are all positive (2 with multiplicity 3), so is positive definite. By the second partial derivative test, this means attains a minimum at this critical point. Meanwhile, has no maximum value.

She should use 12/3 to measure