Consider the following function: z = (x₁ + x₂)² + x₃² + 2x₁ x₃+ 2x₂x₃ .

a. Find the critical points of this function.

b. Does this function reach a maximum or minimum value? (Check for Local Maximum or Minimum) Base on your answer on a discussion of the properties of the Hessian matrix of the function.

c. Find the equation of the tangent plane at the critical points identified by the first order condition.

(Some notes I found which I think are relative to Question c.:

Consider the plane that is just tangent to y = f(x₁,x₂) at the point p where x₁ = x₁* and x₂ = x₂*. Approximating y = f(x) in a neighbourhood of p by reading values off the tangent plane produces . estimates of the change in y moving away from p exactly equal to dy. Thus, the differential of a function is related to its tangent plane.To obtain the equation of the tangent plane you are required to first determine the 2 independent displacement vectors in the plane. This requires 3 points in the plane from which two displacement vectors can be calculated or some other information that will gives two vectors directly.