1. Consider f (x, y,z) = x + y +z subject to the constraints z = 4x + 4y and 2x + 4z=5. (a) (10 pts) Find the gradients of f , and the constraints g = 4x2 + 4y -z and g2 = 2x+4z-5. (b) (20 pts) Write a system of equations using Lagrange multipliers , and 2 whose solution(s) are critical points of the objective function.(c) (40 pts) Find the solutions to the system of (b). List the x ,y - , and z coordinates. Hint: there are two real solutions. (d) (10 pts) Find the maximum and minimum of f (x,y,z) = x2 + y2 + z2 subject to z2 = 43::2 + 4 y2 and 215+ 42 = 5. Note: this is equivalent to nding the maximum and minimum values of f on the curve formed by the intersection of the cone and the plane. 2. (60 pts) Find the points lying on the intersection of x + % y+ %z = 0 and x2 + y2 + z2 = 9 with the largest and smallest z-coordinates. Give exact answers, not approximations