To find the minimum marginal cost, we need to first understand that the marginal cost is the derivative of the cost function, $C(x)$. We’ll follow these steps:

1. Differentiate $C(x)$ with respect to $x$ to find $C'(x)$, which represents the marginal cost.
2. Find the critical points by setting $C'(x) = 0$.
3. Evaluate the marginal cost at these critical points and the boundaries $x = 0$ and $x = 110$ to determine the minimum marginal cost.

Step 1: Differentiate $C(x)$

Given: $C(x) = 0.0001x^3 - 0.018x^2 + 12x + 110$

Differentiating with respect to $x$: $C'(x) = 0.0003x^2 - 0.036x + 12$

Step 2: Set $C'(x) = 0$ to find critical points

Solving: $0.0003x^2 - 0.036x + 12 = 0$

This is a quadratic equation, which we can solve using the quadratic formula: $x = \frac{-(-0.036) \pm \sqrt{(-0.036)^2 - 4 \cdot 0.0003 \cdot 12}}{2 \cdot 0.0003}$

Simplifying: $x = \frac{0.036 \pm \sqrt{0.001296 - 0.0144}}{0.0006}$ $x = \frac{0.036 \pm \sqrt{0.013104}}{0.0006}$

Calculating the values of $x$ from this will provide the critical points.

Step 3: Evaluate $C'(x)$ at critical points and endpoints

Once we have the critical points, we will evaluate $C'(x)$ at these points and also at $x = 0$ and $x = 110$.

Let me calculate this for you.The real values we obtained for the marginal cost at the endpoints are:

• At $x = 0$: $C'(0) = 12.00$
• At $x = 110$: $C'(110) = 11.67$

The critical points yielded complex numbers, which are not within the domain of interest for this problem.

Conclusion

The minimum marginal cost of the product, rounded to the nearest cent, is $11.67 at $x = 110$.

Would you like a further breakdown of any of the steps, or do you have additional questions?

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1. What is the significance of a minimum marginal cost in production?
2. How would the cost function change if an additional fixed cost was added?
3. What does it mean for a function to have complex critical points in this context?
4. How would we interpret marginal cost in an economic or business context?
5. Could there be a scenario where the minimum marginal cost occurs at $x = 0$?

Tip: When differentiating polynomials, each term's exponent decreases by one, making it a straightforward process to find marginal costs in cost functions.