8.4 This problem deals with the function f : 93 -> defined by f (x, y, z) = x2 + 4y3 + 23 + 2xz+ (x2 + yz + z2) cos xyz. Note that 0 = (0, 0, 0) is a critical point of f. (a) Show that q(x, y, z) = 2x2 +5y2 + 2z2 + 2xz is the quadratic form of fat 0 by sub- stituting the expansion cost = 1 - $12 + R(t), collecting all second degree terms, and verifying the appropriate condition on the remainder R(x, y, z). State the uniqueness theorem which you apply. (b) Write down the symmetric 3 x 3 matrix A such that q(x, y, z) = (x y z) Ay By calculating determinants of appropriate submatrices of A, determine the behavior of f at 0. (c) Find the eigenvalues 1, 12, As of q. Let V1, V2, Va be the eigenvectors corresponding to 1, 12, 13 (do not solve for them). Express q in the wow-coordinate system determined by V1, V2, V3. That is, write q(uv, + vv2 + WV3) in terms of u, v, w. Then give a geometric description (or sketch) of the surface 2x2 + 5y2 + 2z2 + 2xz = 1