FindMaxValue[{3/4 * ln (abs[x1]) + 1/4 * ln (abs[y1]) +
   2/3 * ln (abs[x2]) + 1/3 * ln (abs[y2]), px  > 0, py > 0, x1  > 0,
  y1 > 0, x2 > 0, y2 > 0,  x1 + x2 == 3, y1 + y2 == 3,
  px*(x1 - 2) + py*(y1 - 1) == 0,
  px*(x2 - 1) + py*(y2 - 2) == 0}, {x1, y1, x2, y2, px, py}]