Question

The objective function $z=4x_1+5x_2$, subject to $2x_1+x_2\ge 7,2x_1+3x_2\le 15,x_2\le 3,x_1,x_2\ge 0$ has minimum value at the point.

• A. On x-axis
• B. On y-axis
• C. At the origin
• D. On the parallel to x-axis

Answer 1

The correct option is A On x-axis

Value of $z=4x_1+5x_2$

Convert the given inequalities into equalities to get the corner points

$2x_1+x_2=7$ ....... $\left(i\right)$

At $x_1=0,x_2=7$ and $x_2=0,x_1=3.5$

So, the corner points of $\left(i\right)$ are $(0,7)$ and $(3.5,0)$

$2x_1+3x_2=15$ ...... $\left(ii\right)$

At $x_1=0,x_2=5$ and $x_2=0,x_1=7.5$

So, the corner points of $\left(i\right)$ are $(0,5)$ and $(7.5,0)$

$x_2=3$ ...... $\left(iii\right)$

Plot these corner points on the graph paper and the line given in $\left(iii\right)$

The shaded part shows the feasible region.

At $x_2=3,x_1=2$ in $\left(i\right)$ and $x_1=3$ in $\left(ii\right)$

The corner points of the feasible region are $(3.5,0),(7.5,0),(3,3)$ and $(2,3)$

Corner points $Z=4x_2+5x_2$

$(3.5,0)$ $14$

$(7.5,0)$ $30$

$(3,3)$ $27$

$(2,3)$ $23$

Minimum value of $Z=14$ and it lies on $x$-axis.