If
C(x) = 18000 + 400x − 2.2x^2 + 0.004x^3
is the cost function and
p(x) = 2800 − 7x
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.)
To find the production level that maximizes profit, we need to determine the point at which the marginal revenue equals the marginal cost. The marginal revenue is the derivative of the demand function, and the marginal cost is the derivative of the cost function.
1. First, let's find the derivative of the demand function, p(x):
p'(x) = -7
The derivative of a constant is always zero, and the derivative of x is 1. Therefore, the derivative of -7x is -7.
2. Next, let's find the derivative of the cost function, C(x):
C'(x) = 400 - 4.4x + 0.012x^2
To find the derivative, we use the power rule for derivatives. The constant term 18000 disappears when taking the derivative, and each term is differentiated using the power rule.
3. Now, we can set the marginal revenue equal to the marginal cost and solve for x:
-7 = 400 - 4.4x + 0.012x^2
Rearranging the equation:
0.012x^2 - 4.4x + 407 = 0
This is a quadratic equation. To solve it, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 0.012, b = -4.4, and c = 407.
Plugging in the values:
x = (-(-4.4) ± √((-4.4)^2 - 4 * 0.012 * 407)) / (2 * 0.012)
Simplifying further:
x = (4.4 ± √(19.36 - 19.44)) / 0.024
x = (4.4 ± √(-0.08)) / 0.024
Since the square root of a negative number is undefined, it means that there is no real solution. Hence, we conclude that there is no production level at which profit is maximized based on the given cost and demand functions.
profit = demand*price - cost
just a simple polynomial