Assume that f(x,y,z) = x^2 +y^2 +z^2 +xz has a critical point at (0,0,0). Is f(0,0,0) a maximum, minimum, saddle point, or do we not have enough information to determine this?

Question

Show all of your work and every step to receive full credit. If there is not enough information to determine if (0,0,0) is a max, min, or saddle, please justify your answer.

Answer 1

f(x, y, z) = y² + 1/2x² + 1/2z² + 1/2(x + z)² ≥ 0 and because at (0, 0, 0) this function has minimum = 0