The total cost of producing x units is C(q) = 2q3- 6q2-18q+108 (a) At what level of production will the total cost be minimized? (b) At what level of production will the marginal cost be minimized? (c) At what level of production will the average cost be minimized? (use calcu- lator to approximate the solution.)
To solve these questions, we need to find the derivative of the cost function and set it equal to zero. This will give us the critical points, which we can then analyze to determine the minimum values.
(a) To find the level of production at which the total cost is minimized, we need to find the minimum point of the cost function C(q).
Step 1: Calculate the derivative of the cost function C(q) with respect to q.
C'(q) = 6q^2 - 12q - 18
Step 2: Set the derivative equal to zero and solve for q.
6q^2 - 12q - 18 = 0
You can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Here, we will use the quadratic formula.
Step 3: Apply the quadratic formula to find the values of q.
q = (-(-12) ± √((-12)^2 - 4(6)(-18))) / (2(6))
q = (12 ± √(144 + 432)) / 12
q = (12 ± √576) / 12
q = (12 ± 24) / 12
This gives us two possible values for q: q = 4 and q = -1.
Step 4: Determine which value is the minimum point by evaluating the second derivative or testing the values.
To find which one is the minimum, we can evaluate the second derivative of C(q).
C''(q) = 12q - 12
If C''(q) > 0, the point is a minimum.
If C''(q) < 0, the point is a maximum.
Let's evaluate the second derivative for each value of q.
For q = 4:
C''(4) = 12(4) - 12 = 48 - 12 = 36
Since C''(4) > 0, the cost is minimized at q = 4.
For q = -1:
C''(-1) = 12(-1) - 12 = -12 - 12 = -24
Since C''(-1) < 0, the cost is not minimized at q = -1.
Therefore, the level of production at which the total cost is minimized is q = 4.
(b) To find the level of production at which the marginal cost is minimized, we need to find the minimum point of the derivative function, C'(q).
Step 1: Calculate the derivative of the cost function's derivative, C''(q).
C''(q) = 12q - 12
Step 2: Set the derivative equal to zero and solve for q.
12q - 12 = 0
12q = 12
q = 1
Therefore, the level of production at which the marginal cost is minimized is q = 1.
(c) To find the level of production at which the average cost is minimized, we need to find the minimum point of the average cost function, which is given by C(q)/q.
Step 1: Calculate the average cost function, AC(q) = C(q)/q.
AC(q) = (2q^3 - 6q^2 - 18q + 108) / q
AC(q) = 2q^2 - 6q - 18 + (108 / q)
Step 2: Calculate the derivative of the average cost function, AC'(q).
AC'(q) = 4q - 6 - (108 / q^2)
Step 3: Set the derivative equal to zero and solve for q.
4q - 6 - (108 / q^2) = 0
In this case, using a calculator or numerical approximation methods may be more practical to find the solution.
You can plug the equation into a graphing calculator, a numerical solver, or use iterative methods to find the root of the equation. The solution will give you the level of production at which the average cost is minimized.