A certain cost function has the following graph.

On the coordinate plane the horizontal axis is labeled Items and the vertical axis is labeled Dollars. The curve enters the window at Dollars = 400 on the positive Dollars axis, goes up and right becoming less steep, passes through the point (50, 1200), passes through the approximate point (83, 1470), goes up and right becoming more steep, and exits the window in the first quadrant.

(a)

The associated marginal cost is  ---Select--- increasing, then decreasing decreasing, then increasing always decreasing always increasing .

(b)

The marginal cost is least at approximately  ---Select--- x = 0 x = 50 x = 83 x = 150 .

(c)

The cost of 50 items is  ---Select--- approximately $1.6, and increasing at a rate of about $1,200 per item approximately $1,200, and increasing at a rate of about $1.6 per item approximately $1,200, and increasing at a rate of about $8 per item approximately $8, and increasing at a rate of about $1,200 per item .

Let's analyze the description of the cost function graph in each part.

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### (a) The associated marginal cost

The marginal cost refers to the rate at which the cost increases with the number of items, which is essentially the slope of the cost function. From the graph's description:
- The curve **initially becomes less steep**, which suggests that the rate of cost increase (marginal cost) is **decreasing**.
- The curve then **becomes more steep**, which suggests that the rate of cost increase (marginal cost) is **increasing**.

Thus, the marginal cost first decreases and then increases. Therefore, the correct answer is:

**(a) The associated marginal cost is: decreasing, then increasing.**

---

### (b) The marginal cost is least at approximately:

The marginal cost is least where the slope of the graph is the lowest, which is when the curve is flattest. Based on the description, the curve becomes less steep up to around **(50, 1200)** and then starts increasing in steepness after that point.

Thus, the marginal cost is likely least at around **x = 50**.

**(b) The marginal cost is least at approximately: x = 50.**

---

### (c) The cost of 50 items

From the description, the graph passes through the point **(50, 1200)**, meaning that the cost of 50 items is approximately $1200.

Next, we are asked about the rate of change of the cost, or the marginal cost, at 50 items. Since the marginal cost refers to how quickly the cost is increasing, the graph describes that around x = 50, the curve is getting flatter, and the cost is increasing more slowly. The marginal cost at that point is approximately **$8 per item**, based on the general behavior of the graph.

Thus, the correct answer is:

**(c) The cost of 50 items is: approximately $1,200, and increasing at a rate of about $8 per item.**

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### Summary:
(a) The associated marginal cost is **decreasing, then increasing**.  
(b) The marginal cost is least at approximately **x = 50**.  
(c) The cost of 50 items is **approximately $1,200, and increasing at a rate of about $8 per item**.

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Do you want more details or have any questions?  

Here are 5 related questions to explore further:
1. What is the definition of a marginal cost function?
2. How does a cost function typically behave when economies of scale are present?
3. How do you find the marginal cost function from the total cost function?
4. Can the marginal cost ever be negative, and what does that imply in practical terms?
5. How does the shape of a cost function help determine the efficiency of production?

**Tip:** The marginal cost is the derivative of the cost function and gives insight into how quickly costs change as more items are produced.

A certain cost function has the following graph.

On the coordinate plane the horizontal axis is labeled Items and the vertical axis is labeled Dollars. The curve enters the window at Dollars = 400 on the positive Dollars axis, goes up and right becoming less steep, passes through the point (50, 1200), passes through the approximate point (83, 1470), goes up and right becoming more steep, and exits the window in the first quadrant.

(a) The associated marginal cost is ---Select--- increasing, then decreasing decreasing, then increasing always decreasing always increasing.

(b) The marginal cost is least at approximately ---Select--- x = 0 x = 50 x = 83 x = 150.

(c) The cost of 50 items is ---Select--- approximately $1.6, and increasing at a rate of about $1,200 per item approximately $1,200, and increasing at a rate of about $1.6 per item approximately $1,200, and increasing at a rate of about $8 per item approximately $8, and increasing at a rate of about $1,200 per item.

Explore the behavior of a cost function with marginal cost analysis, including decreasing then increasing cost rates. This example examines key points such as the cost at 50 items being $1200 and the marginal cost rate of $8 per item. Suitable for high school students in Grades 10-12.

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