@abhinash,

Hi.

Firstly, as you can notice, the TC function is a parabola (try and draw the graphics if you don't believe me).

Therefore, you can get the function of the parabola by setting a system of equations.

The total cost will be like this: TC = aQ^2+ bQ+c, where Q is the quantity produced.

• 10 = a (0)^2+b(0)+c

• 12 = a(1)^2+b(1)+c

• 16 = a(2)^2+b(2)+c

______________

• c = 10

• 12 = a+b+10

• 16 = 4a+2b+10 (then I'll divide it by two)

______________

• c = 10

• 2 = a+b

• 8-5 = 2a+b

______________

• c = 10

• a+b = 2

• 3= 2a+b

______________

• c= 10

• a = 2-b

• 3 = 2 (2-b) + b

______________

• c= 10

• a = 2-b

• 3 = 4-2b+b

______________

• c = 10

• a = 2-b

• 3-4 =-b

_______________

• c= 10

• a = 2-b

• b = 1

_______________

• c = 10

• a = 1

• b = 1

TC = Q^2+Q+10

Now, the profit is maximized when MC (Marginal Cost) equals MR,

(Marginal Revenues), which is equal to the market Price.

• If P>MC the profit increases by P-MC by producing one more unit

• If P<MC, the profit diminishes by P-MC (which is negative), but more importantly, by decreasing the quantity produced, the MC diminishes faster than the price, and the profit increases.

• If P=MC the profit could be maximum or minimum. It is maximum when the MC is increasing.

MC = 2Q+1 ( = dTC/dQ)

The total revenue function is a straight line whose equation is: TR = 7Q

MR = 7 = P

2Q+1 = 7

2Q = 6

Q = 6/2

Q = 3

NOW WE CHECK THAT THE MC IS INCREASING.

The parabola in convex, and the minimum can be found in two different ways:

-b/2a

= -1/2(1) = -1/2

OR

2Q+1>0

2Q > -1

Q > -1/2

For Q>-1/2 the parabola in increasing

If Q<0 there's no economic interpretation, so you have to consider a restriction of the domain to (0, + infinity), so the solution we found (Max profit --> Q=3) is a possible solution.

The economic profit can be found as TR-TC(Q), and it is 7*3-22 = 21-22 = -1

*NOTICE THAT* the economic profit is always negative for every level of Q.

However, if the firm didn't produce, it'd have to pay 10€ (or $) for she total sunk fixed costs, having a profit of -10. The profit of -1 is still a loss, but the firm loses less than by producing 0.

I don't know how to use the integers so I'll leave it to those who can do it.