1. The paraboloid z = :32 + 3,12 intersects the sphere of radius 3 centred at (3, 0, 4) along a curve passing through the point P = (1, 2, 5). (a) Find a normal vector to the paraboloid at the point P. (b) Find a normal vector to the sphere at the point P. (c) Find a tangent vector to the curve of intersection at the point P. [2+2+2 marks] 2. Let f(:r,y) =xQny+2y2:r+3. (a) Determine all the critical points of f. (b) Determine the nature of each critical point of f. (c) Determine the maximum and minimum of f on the boundary of the triangle with verticals (0,0), (2, 0) and (1,1). (d) Determine the absolute maximum and minimum of f over the entire triangle with vertices (0,0), (2, 0) and (1,1). [2+2+4+1 marks] 3. Use the method of Lagrange multipliers to nd the minimum distance between the paraboloid z = 11:2 + 433 and the point (3,0, 0). [6 marks] 4. (a) Use the method of Lagrange multipliers to show that if 2:1 +x2+cc3 = 1 and 1:1, 1:2, x3 2 0, then 1 3:11:32 + 311:3 + 323:3 5 E