State the Karush–Kuhn–Tucker optimality conditions for each of the following mathematical programs.

(a) min 141x1 - 922 + 31x2 - 522 + 1x3 - 1122 s.t. 2x1 + 18x2 - x3 = 19 6x1 + 8x2 + 3x3 … 20 x1, x2 Ú 0

(b) max 6x1 + 40x2 + 5x3 s.t. x1 sin1x22 + 9x3 Ú 2 e18x1 + 3x2 + 14x3 … 50 x2, x3 Ú 0

(c) min 100 - 1x1 - 322 - 1x224 + 19x3 s.t. 51x1 - 122 + 301x2 - 222 Ú 35 60x2 + 39x3 = 159 x1, x2, x3 Ú 0

(d) max 7 ln1x12 + 4 ln1x22 + 11 ln1x32 s.t. 1x1 + 222 - x1x2 + 1x2 - 722 Ú 80 5x1 + 7x3 Ú 22 x1, x2, x3 Ú 3