If C(x) = 13000 + 600x − 1.8x^2 + 0.004x^3 is the cost function and
p(x) = 4200 − 6x is the demand function, find the level of output that maximizes utility. (Hint: If utility is maximized, then marginal revenue equals marginal cost.)
define "utility", not familiar with this kind of math
To find the level of output that maximizes utility, we need to find the output level where the marginal revenue equals the marginal cost. The marginal revenue is the derivative of the demand function p(x), and the marginal cost is the derivative of the cost function C(x).
Step 1: Find the equation for marginal revenue (MR).
MR = p'(x)
Given: p(x) = 4200 - 6x
Take the derivative with respect to x:
p'(x) = -6
So, the equation for marginal revenue is MR = -6.
Step 2: Find the equation for marginal cost (MC).
MC = C'(x)
Given: C(x) = 13000 + 600x - 1.8x^2 + 0.004x^3
Take the derivative with respect to x:
C'(x) = 600 - 3.6x + 0.012x^2
So, the equation for marginal cost is MC = 600 - 3.6x + 0.012x^2.
Step 3: Set MR equal to MC and solve for x.
-6 = 600 - 3.6x + 0.012x^2
This equation is a quadratic equation. Rearrange the equation to make it equal to zero:
0.012x^2 - 3.6x + 606 = 0
Now, solve this quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
a = 0.012, b = -3.6, c = 606
x = (3.6 ± √((-3.6)^2 - 4 * 0.012 * 606)) / (2 * 0.012)
Simplifying the expression inside the square root:
x = (3.6 ± √(12.96 - 2904.48)) / 0.024
x = (3.6 ± √(-2891.52)) / 0.024
Since the square root of a negative number is not real, it means there is no real solution for x. This means there is no level of output that maximizes utility according to the given functions.