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2015-04-02T02:59:34+05:30

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The vertices are not given in the order.
A = (0, 0);    B (12, 0)  ;      C(12, 5) ;      D(0, 5)

We need to evaluate the value of Q at the vertices , on the boundaries of lines AB, BC, CD, DA and inside the rectangle given by the constraint equations.

The equations corresponding to the sides of the rectangle ABCD:
       AB : (y - 0)/(x-0) = 0/12            => y = 0
       BC :  x = 12
       CD :  y = 5
       DA:   x = 0

The region inside and on the sides of the rectangle is given by the constraints:
           0 ≤ x ≤ 12      and    0 ≤ y ≤ 5

a)
     Q = 7 x + 14 y
         as the values of x and y are non-negative, it is easy to see the maximum and minimum values of Q.
         Minimum value of Q  at A ( 0, 0) :   0
         Maximum value of Q  at C (12, 5) :  7 *12 + 14 * 5 = 154
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b)  we know x and y are non-negative.

       Q = - 9 x + 20 y
     It is easy to see that Q is maximum when y is maximum and x is minimum.
                Max value of Q at D(0, 5) :  - 9 * 0 + 20 * 5 = 100

     Also, Q is minimum when x is maximum and y is minimum.
           MAx value of Q at B (12, 0):    -9 * 12 + 20 * 0 = -108.

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