Who is asking: Student
The marginal cost for a certain product is given by MC = 6x+60 and the fixed costs are $100. The marginal revenue is given by MR = 180-2x. Find the level of production that will maximize profit and find the profit or loss at that level.
If C(x) is the cost function where x is the level of production of the product being produced them the marginal cost, MC(x), is the derivative of C(x). Thus, using antidifferentiation you can find the cost function. Since
MC(x) = C'(x) = 6 x + 60
C(x) = 3 x2 + 60 x + K
where K is some constant.
You are also told that the the fixed costs are $100, in other words C(0) = $100. Thus
C(0) = 3(0)2 + 60(0) + K = 100
Thus K = 100 and hence
C(x) = 3 x2 + 60 x + 100
In a similar way you can find the revenue function, M(x), since M'(x) = MR(x). In this case, if you produce no product the revenue is $0 and hence R(0) = 0.
The profit function, P(x), is the revenue the cost, and hence
P(x) = R(x) - C(x).
Thus to maximize the profit, find P'(x) and solve P'(x) = 0. Notice that the equation you have to solve is
MC(x) = MR(x)
I hope this helps,
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