To determine the marginal cost function, we first need to understand what the marginal cost is. The marginal cost is the derivative of the total cost function with respect to the number of units produced. In other words, it represents the rate at which the total cost changes as the production level changes.
Given the total cost function:
[tex]\[ C(x) = 2x^3 - 20x^2 + 160x + 4000 \][/tex]
We need to find the derivative of [tex]\( C(x) \)[/tex] with respect to [tex]\( x \)[/tex]. Let's differentiate each term of the function step by step:
1. Differentiate [tex]\( 2x^3 \)[/tex]:
[tex]\[ \frac{d}{dx}(2x^3) = 3 \cdot 2x^{3-1} = 6x^2 \][/tex]
2. Differentiate [tex]\( -20x^2 \)[/tex]:
[tex]\[ \frac{d}{dx}(-20x^2) = 2 \cdot (-20)x^{2-1} = -40x \][/tex]
3. Differentiate [tex]\( 160x \)[/tex]:
[tex]\[ \frac{d}{dx}(160x) = 160 \][/tex]
4. The derivative of the constant term [tex]\( 4000 \)[/tex] is:
[tex]\[ \frac{d}{dx}(4000) = 0 \][/tex]
Combining all these results, we get the marginal cost function [tex]\( MC(x) \)[/tex]:
[tex]\[ MC(x) = 6x^2 - 40x + 160 \][/tex]
Therefore, the correct marginal cost function is:
[tex]\[ \boxed{6x^2 - 40x + 160} \][/tex]
Among the provided options, the correct answer is:
c. [tex]\( 6x^2 - 40x + 160 \)[/tex]
