Let the corner points of the bounded feasible region of an LPP determined by a set of constraints are A(1, 5), B(3, 2) and C(2, 1). If the objective function \( z=2ax+by,\ a,b\gt 0 \) has maximum values at points A and B and z has minimum value 8 at point C, then the value of \( 4a+b \) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 8
The value of z at C(2, 1) is
\( z(C)=2a(2)+b(1)=4a+b \)
The minimum value at C is 8, so
\( 4a+b=8 \)
Check with the condition at A and B: z(A) = z(B) gives \( 2a+5b=6a+2b \), so \( 3b=4a \). With \( 4a+b=8 \) this gives b = 2 and a = 3/2; then z(A) = z(B) = 13 and z(C) = 8, so z is indeed largest at A and B and smallest at C.
Tip: Here \( 4a+b \) is exactly z at C, so no solving is needed; 13, Option (4), is the maximum value of z, not \( 4a+b \).