Find the maximum and minimum values of f(x,y,z) = x^2 - 2x + y^2 - 4y + z^2 - 4z in the region x^2 + y^2 + z^2 <= 36.
The gradient of f = lambda * gradient of g
<Fx, Fy, Fz> = lambda<Gx, Gy, Gz>
Test for local extrema
If Determinant D(x,y) < 0 it is a saddle point
If D(x,y) > 0:
If Fxx > 0, then it is a local minimum
If Fxx < 0, then it's a local maximum
The Attempt at a Solution
Tried to solve for system of equations that result in 4 equations[/B]
2x-2 = lambda * 2x
2y-4 = lambda * 2y
2z-4 = lambda * 2z
X^2 + y^2 + z^2 = 36
I solved for lambda = 1/2 or 3/2 by solving for x y and z and substituting into the 4th equation.
I got (2,4,4) =0 and (-2,-4,-4) =72. BUT the minimum was actually (1,2,2) = -9 because I was supposed to "check the interior of the sphere". Am I just supposed to account for the possibility that lambda = 0?