conservative and optimization. Part c + d

For the next problems, let M be the ball of radius centered at the origin of R3, that is, M: {(w,y,z) : :32 +192 +2!2 S 3}. Consider a particle that moves in M and is subject to the force eld given by F(m,y, z) = (2mg, 1:2, 22). (3) Check thatFis a conservative eld by showing thatV x F = 0. Find a potential for F,that is, a scalar function ; 3 such that f R > R Vf = F, and use this potential function to show that the work done by Fto move our particle from (1, 1, 1) to a point (:3, y, z) in M is given by W(m,y, z) = a22y+ 22 2. (b) Let'snd 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 MA Show that a critical point ofWin i1:Ll:(M)={(:)3,y,z):a:2 +112 +22