Question

Given the cost function C(x) = 3x3 - 2x² + 2x + 3, find the minimum marginal cost. The minimum marginal cost is $0 (Do not round until the final answer. Then round to two decimal places as needed.)

Answer 1

The minimum marginal cost is $0.

Step-by-step explanation:

The minimum marginal cost can be found by taking the derivative of the cost function with respect to x and setting it equal to 0. Let's find the derivative first:

C'(x) = 9x^2 - 4x + 2

Setting C'(x) = 0:

9x^2 - 4x + 2 = 0

Using the quadratic formula to solve for x:

x = (-(-4) ± sqrt((-4)^2 - 4*9*2)) / (2*9)

x = (4 ± sqrt(16 - 72)) / 18

x = (4 ± sqrt(-56)) / 18

Since taking the square root of a negative number results in a complex number, there are no real solutions for x. Therefore, there is no minimum marginal cost.

Answer 2

To find the minimum marginal cost, calculate the derivative of the cost function and solve for x when it equals zero. However, since the quadratic equation has no real solutions, there is no minimum marginal cost.

To find the minimum marginal cost, we need to calculate the derivative of the cost function and find where it equals zero. Let's first find the derivative of the given cost function C(x) = 3x³ - 2x² + 2x + 3:

C'(x) = 9x² - 4x + 2

Next, set C'(x) = 0 and solve for x:

9x² - 4x + 2 = 0

Using the quadratic formula, we get:

x = (-(-4) ± √((-4)² - 4(9)(2))) / (2(9))

x = (4 ± √(16 - 72)) / 18

x = (4 ± √(-56)) / 18

Since the square root of a negative number is not real, there are no real solutions. Therefore, the minimum marginal cost is not possible.