1. (44-44-444 points) Let z, y) be a function whose partial derivatives of all orders exist every- where (meaning fm, fy, fm, fw, fan\". . . all exist). For each of the following statements, decide whether it is true or false, and briey (but rigorously!) justify your answers. Note: Although it is not explicitly stated in the course notes, you may assume that all local extrema of a function must occur at critical points. (a) If ((1,6) and (rid) are two points such that u, b) = c, til), then if TL} is the unit vector pointing from ((1,5) to (ad), we have Dal\": b) = II]. (b) If ((1,3)) is such that fm(a,b) 99 {I and f\"(e,b) = fw(a,b), then [(1,6) is not a local extremum for f(z,y). (c) If 9(3, 3;) is another function whose partial derivatives of all orders exist, then all absolute extrema of f(z,y) subject to a constraint of the form g{z,y) = K occur when Vf = AVg for some real number A. (d) Suppose that I and y are each a function of s and t, and that % = aim and $3 = g\". Then 3f 3)" 333t