3. (This problem has two parts and continues on the next page.) Suppose z = f(x, y) is a continuous function with level curves PP The labels on the level curves are the values of z at which they occur. Assume that other level curves occur at values of z strictly between those that are drawn. The r-axis (not drawn) passes through (0,0) horizontally, and the y-axis is the level curve drawn at z = 4. (a) (10 points) Assuming that fr and fy exist at all points, determine whether each of the following is positive, negative, or zero. fx(Q), fu(Q), fx(0, 0), fu(0, 0), fx(P), fu(P) Justify your answer for fr(0, 0) and fy(0,0) only. fx ( Q ) so fy ( Q) so fx (P) b fx 1010) = 0 since f is at a maximum at (o,0 ) in the X-direction because f ( olo ) 2 f ( X(o) for x close to O fyloo) = 6 since of is constant in the y- direction at ( 6.0 )(b) (15 points) Determine, with appropriate justification, the value of the following limit. lim (f(x, y ) + 20' + 3xy (x,y)-(0,0) VI2 + yz Since f is continuous lim f ( x,y ) = f 1010) = 4. (Xiy1 - (0.0) For lim 2x + 3xy convert to polar ( X 141 - 10. 6) coordinates Get lim 2 n2 cos20 + 3x2 coso sino r = lim - [ 2 cos * 0 + 3 coso sing Since - 5r ( 610)