The goal of this quiz is to see if you are comfortable with conservative elds, but it is also a review of optimization. For the next problems, let M be the ball of radius 2 centered at the origin of R3, that is, M: {($,y,z) : :32 +19!2 +22 :4}. Consider a particle that moves in M and is subject to the force eld given by Flay-l2) = (293, 292, 92)- (a) Check that F is a conservative eld by showing that V X F = 0. Find a potential for F, that is, a scalar function f : R3 } R such that Vf = F, and use this potential function to show that the work done by F to move our particle from (1, 1, D) to a point (x, y, z) in M is given by Whoa/'1): 2922 :62 + 1- (b) Let's find the maximum and minimum values of W. To answer this question, recall that we consider two types of candidates for max/min: some are values attained at points in the interior of M, and some are values attained at points in the boundary of M. Show that a critical point of W in int (M) = {(x, y, z) : 32 + 2 + 22