If \( Z=3x+4y \) is the objective function subject to the constraints \( 3x+2y\le 6,6x+y\le 6,x\ge 0,y\ge 0 \) such that the corner points of the bounded feasible region are (0, 0), (\( a+3 \),0), (0, \( b-1 \)) and (\( c,d \)). The value of Z at \( \left(-\displaystyle\frac{1}{c}-a,b-d\right) \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{19}{2} \)
The corner points of the feasible region lie on the axes and at the meeting point of the two lines.
On the x-axis the binding constraint is \( 6x+y\le 6 \), giving (1, 0) (the other line meets the axis at (2, 0), which is outside). On the y-axis it is \( 3x+2y\le 6 \), giving (0, 3) (the other line meets it at (0, 6), outside). The fourth corner is where the two lines meet:
\( \displaystyle 6x+y=6\ \Rightarrow\ y=6-6x \)
\( \displaystyle 3x+2(6-6x)=6\ \Rightarrow\ -9x=-6 \)
\( \displaystyle x=\frac{2}{3},\ y=2 \)
Comparing with the given corners:
\( \displaystyle a+3=1\ \Rightarrow\ a=-2 \)
\( \displaystyle b-1=3\ \Rightarrow\ b=4 \)
\( \displaystyle c=\frac{2}{3},\ d=2 \)
The required point is
\( \displaystyle \left(-\frac{1}{c}-a,\ b-d\right) \)
\( \displaystyle =\left(-\frac{3}{2}+2,\ 4-2\right)=\left(\frac{1}{2},\ 2\right) \)
\( \displaystyle Z=3\left(\frac{1}{2}\right)+4(2) \)
\( \displaystyle =\frac{3}{2}+8=\frac{19}{2} \)