Question

Given the cost function C(x)=5x³−4x²/2x−8, find the minimum marginal cost:

a) Use the first derivative test
b) Set the second derivative equal to zero
c) Find the critical points
d) None of the above

Answer 1

The minimum marginal cost can be found using the first derivative test, finding critical points where the first derivative equals zero, and confirming these are minima with a positive second derivative.

Step-by-step explanation:

To find the minimum marginal cost using the cost function C(x)=5x³−4x²/2x−8, follow these steps:

1. Use the first derivative test: First, differentiate the cost function to get the marginal cost, which is the derivative of the cost function C'(x). Then, find where this derivative equals zero to locate potential minima.
2. Set the second derivative equal to zero: This method is incorrect for finding minima. Instead, compute the second derivative and assess whether it is positive at the critical points found in step one to confirm if they are indeed minima.
3. Find the critical points: Solve the first derivative equation for zero to find the critical points where the marginal cost could have a minimum.
4. None of the above: This is not a valid method for finding the minimum of a function.

The correct approach is to use the first derivative test along with evaluating the second derivative at critical points (not setting it to zero) to determine where the function has minimum points.