Question

a) Minimize and maximize $Z=x+2y$, subject to the constraints
$x+2y\ge 100$
$2x-y\le 0$
$2x+y\le 200$
$x,y\ge 0$ by graphical method.
b) Prove that $\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|=4abc$

Answer 1

$\left(a\right)$

$Z=x+2y$, subject to the constraints
$x+2y\ge 100$
$2x-y\le 0$
$2x+y\le 200$
$x-y\ge 0$ by graphical method.

On solving equations $2x-y=0$ and $x+2y=100$ we get point $B(20,40)$

On solving $2x-y=0$ and $2x+y=200$ we get $C(50,100)$

$\therefore$ Feasible region is shown by $ABCDA$

The corner points of the feasible region are $A(0,50),B(20,40),C(50,100),D(0,200)$

Let us evaluate the objective function $Z$ at each corner points as shown below

At $A(0,50),$ $Z=0+100=100$

At $B(20,40),Z=20+80=100$

At $C(50,100),Z=50+200=250$

At $D\left(0.200\right),Z=0+400=400$

Hence, Maximum value of $Z$ is $400$ at $D(0,200)$ and minimum value of $Z$ is $100$ at $A$ and $B$.

$\left(b\right)$

Consider, $\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|$

$=(b+c)\left|\begin{array}{cc}c+a& b\\ c& a+b\end{array}\right|-a\left|\begin{array}{cc}b& b\\ c& a+b\end{array}\right|+a\left|\begin{array}{cc}b& c+a\\ c& c\end{array}\right|$

$=(b+c)\left[\right(c+a\left)\right(a+b)-bc]-a[ab+b^2-bc]+a[bc-c^2-ac]$

$=(b+c)a(b+c+a)-a\left[b\right(b-c+a\left)\right]+a[-c(-b+c+a\left)\right]$

$=4abc$

Hence, $\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|=4abc$