a) The business firm produces only two products A and B. It is given that x represents the amount of product A and y represents the amount of product B.

Now, the firm is required to produce total 35 units of x and y. i.e. x + y = 35

The firms main objective should be minimizing its cost to produce x and y in such a way that the requirement of 35 units gets fulfilled.

The logical explanation is that the firm wants to maximize its profit.

Profit = Total Revenue - Total Cost

As the total cost gets minimized, automatically the profit is maximized given total revenue is fixed. So, in this situation it is best interest of the firm that it has to minimise its cost of production.

b) The firm will be assessing the problem as:

Min C = 5x² + 15y²

s.t. x + y =35

The firm has a constraint of producing total 35 units of product A and product B.

The above minimisation problem can be solved using Lagrange optimization method:

Z = 5x² + 15y² + λ(35 - x - y)

The first order conditions by partial differentiation are as follows:

i) dZ/dx = 10x - λ = 0

             => 10x = λ ----------(1)

ii) dZ/dy = 30y - λ = 0

              => 30y = λ    ----------(2)

iii) dZ/dλ = 35 - x - y = 0

              => x + y = 35   -------------(3)

Now, comparing equation (1) and equation (2) we get -

10x = 30y

                                                                                 => x = (30/10)y  

                                                                                       => x = 3y                -------(4)

Now, putting the value of x in equation (3), we get -

x + y = 35

=> 3y + y = 35

=> 4y = 35

=> y = 35/4

=> y = 8.75

Putting the value of y to get x -

x = 3y

=> x = 3 x 8.75

=> x = 26.25

The firm needs to produce 26.25 amount of product A and 8.75 amount of product B to minimize its cost of production and maximize its profit.