X and Y are two positive real numbers such that 2X + Y = 3 and X + 4Y = 6. For which of the following values of (X, Y) will the function g(X, Y) = 3X + 4Y give the maximum value? A. (4/3, 1/3) B. (8/3, 1/3) C. (8/3, 1/3) [see options] D. (4/3, 20/3)

GATE 2013 · General Aptitude · Maxima Minima · medium

Answer: The maximum of g(X,Y) = 3X+4Y subject to the given constraints is achieved at (X,Y) = (4/3, 1/3), giving g = 16/3.

1. Solve 2X+Y=3 and X+4Y=6 simultaneously: X + 4(3-2X) = 6 => X + 12 - 8X = 6 => -7X = -6 => X = 6/7. Then Y = 3 - 12/7 = 9/7. Intersection: (6/7, 9/7).
2. Identify feasible vertices from each constraint with positivity: Feasible vertices (X>0,Y>0): (3/2, 0) from first constraint x-axis, (0,3) from first y-axis, (6,0) from second x-axis, (0, 3/2) from second y-axis, plus (6/7, 9/7) at intersection. The answer options include (4/3, 1/3): g = 3(4/3)+4(1/3) = 4 + 4/3 = 16/3 ≈ 5.33. Option A gives the maximum among the listed choices.