For each of the following combinations of objective function, point, and direction, determine whether conditions 3.21 and 3.22 show that the direction improves at the point, does not improve at the point, or that further information is required.

(a) max 4y1 - 2y3 + y5, y = 11, 0, 19, 4, 62,

y = 12, -3, 4, 0, 62

(b) max y1 + 7y3 + 2y5, y = 11, 0, 9, 0, 02,

y = 1 -10, -20, 2, 0, 42

(c) min y1y2 + 1y122 + 4y2, y = 13, -12,

y = 1 -7, 52

(d) min y1y2 + 4y1 + 1y222, y = 12, 12,

y = 1 -1, 32

(e) max 1y1 - 522 + 1y2 + 122, y = 14, 12 ,

y = 1 -1, 22

(f) min 1y1 - 222 + y1y2 + 1y2 - 322, y = 11, 12, y = 13, -12