Question

The corner points of the feasible region for optimization problem determined by a system of linear constraints are $(6,5),(9,7),(15,0),(0,10).$ Let $z=px+qy$ and the maximum of $z$ occurs at both the points $(6,5)$ and $(9,7),$ then which of the following is true:

• A. $3p+2q=0$ and $p>0,q>0$
• B. $3p+2q=0$ and $p>0,q<0$
• C. $3p+2q=0$ and $p<0,q<0$
• D. $3p+2q=0$ and $p<0,q>0$

Answer 1

The correct option is C $3p+2q=0$ and $p<0,q<0$

Let $z_0$ be the maximum value of $z$ in the feasible region. Since maximum occurs at both $(6,5)$ and $(9,7)$, the value $z_0$​ is attained at both $(6,5)$ and $(9,7)$.
$\Rightarrow z_0=p\left(6\right)+q\left(5\right)\cdots \left(i\right)$
and $z_0=p\left(9\right)+q\left(7\right)\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right),$ we get:
$6p+5q=9p+7q$
$\Rightarrow 3p+2q=0\cdots \left(iii\right)$
and the values at other corner points $(15,0)$ and $(0,10)$ should be less than the $z_0$
$\Rightarrow 6p+5q>15p\Rightarrow p<0\cdots \left(A\right)\left[\text{from}\right(iii\left)\right]$
and $6p+5q>10q\Rightarrow q<0\cdots \left(B\right)\left[\text{from}\right(iii\left)\right]$