# (Solved): In this question, we seek to understand the stationary points of the function f: R3 R of three vari ...

In this question, we seek to understand the stationary points of the function f: R3 R of three variables given by:f(x, y, z) = x3 - 12x + y3 + 23 - 3yzIn this question, some marks will be awarded for the answer, and some marks for the correct method.(a) [2 . First of all, calculate the stationary points of f. [You should find there are four of them, one of which is at (2, 1, 1).](b). The nature of the stationary points of f can, as in two dimensions, be understoodfrom the Hessian matrix. We will focus on the stationary point a = (2,1, 1). Compute the matrixQ = Hf(2,1,1), and hence write down the best quadratic approximation to f near the point a =(2,1, 1). (You should leave your answer as a polynomial in the variables (x-2), (y-1) and (z-1).)(c) |2 marks). Our goal now is to show that the stationary point (2, 1, 1) is a local minimum of f. We will do so by analysing the matrix Q further. Begin by showing that Q has eigenvalues 3, 9 and 12, and find associated eigenvectors U,, U, and v3.(d) . We will now use the above facts to show that (2, 1, 1) is a local minimum of f.(i) First show that, if A is a 3 x 3 matrix, and v is an eigenvector of A with eigenvalue 2, and w is any vector in R3, then wAv = 2(w v).

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