Question

If C(x) = 16000 + 400x − 0.6x² + 0.004x³ is the cost function and p(x) = 1600 − 6x is the demand function, find the production level that will maximize profit. (Hint: If the profit is maximized, then the marginal revenue equals the marginal cost.)

Answer 1

To maximize profit, the production level should be set where marginal revenue (MR) equals marginal cost (MC). This is achieved by differentiating the revenue function and the cost function, setting these derivatives equal to each other, and solving for the quantity produced (x).

Step-by-step explanation:

To find the production level that maximizes profit, we need to equate marginal cost (MC) and marginal revenue (MR). The cost function C(x) is given by C(x) = 16000 + 400x − 0.6x² + 0.004x³, and the demand function p(x) is given by p(x) = 1600 − 6x. The revenue function R(x) can be found by multiplying the demand function by x, which is R(x) = x * p(x). To find MR, we compute the derivative of the revenue function concerning x, MR = d(R(x))/dx. Similarly, to find MC, we take the derivative of the cost function, MC = d(C(x))/dx. The profit-maximizing output level occurs where MR = MC. By finding these derivatives and setting them equal to each other, we can solve for the value of x that maximizes profit.

The step-by-step process involves:

1. Finding the revenue function R(x) by multiplying the demand function by x.
2. Computing the derivative of R(x) to get MR.
3. Computing the derivative of C(x) to get MC.
4. Equate MR and MC and solve for x.

This process helps us identify the level of production that maximizes profits by aligning the incentive to produce with the costs of production.