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The problem you're working on involves interpreting a graph showing the revenues of two department stores: a discount store and an established store. The two functions, \(f(t)\) for the discount store and \(g(t)\) for the established store, represent their revenues over time \(t\) (in months). 

From the graph, you have three key points:
- \(t_1\), \(t_2\), and \(t_3\) represent specific times where notable changes occur in the revenues.

Now, let's address each part of the question:

### (a) The revenue of the established department store is decreasing at the slowest rate.
To determine this, you need to find where the derivative \(g'(t)\) (the rate of change of \(g(t)\), representing the revenue of the established store) is closest to zero but still negative. This would occur at the point where the graph of \(g(t)\) has the least steep negative slope.

Based on the graph:
- The curve is steepest around \(t_2\), meaning the store is losing revenue quickly.
- The curve flattens before \(t_3\), meaning the store's revenue loss is slowing down.

Thus, the slowest rate of revenue decrease for the established department store likely happens near \(t_3\).

### (b) The revenue of the established department store is decreasing at the fastest rate.
The fastest rate corresponds to the steepest downward slope of \(g(t)\), which happens around \(t_2\) since the curve is steepest at that point.

So, the fastest revenue decrease occurs at \(t_2\).

### (c) The revenue of the discount store first overtakes that of the established store.
This is the point where the graph of \(f(t)\) (the revenue of the discount store) intersects and surpasses the graph of \(g(t)\). From the graph, this happens at \(t_2\), where the two curves intersect.

### Answers Summary:
- (a) Slowest rate: \(t_3\)
- (b) Fastest rate: \(t_2\)
- (c) Revenue overtakes: \(t_2\)

Would you like a more detailed explanation of any of these points? Here are five follow-up questions you can explore:

1. How would you calculate the exact rate of change of revenue for each store at any given time?
2. What mathematical tools could be used to find the exact intersection point \(t_2\)?
3. Can you think of a real-world scenario where such curves might apply?
4. How does concavity relate to the speed of revenue changes in this scenario?
5. How could this model be extended to predict future revenues?

**Tip:** The slope of the curve at any point gives valuable information about the rate of change. If you’re asked about the slowest or fastest rate of change, always examine the steepness of the curve at different points.

The following figure shows the devastating effect the opening of a new discount department store had on an established department store in a small town. The revenue of the discount store at time t (in months) is given by f(t) million dollars, whereas the revenue of the established department store at time t is given by g(t) million dollars. Answer the following questions by giving the value of t at which the specified event took place: (a) The revenue of the established department store is decreasing at the slowest rate. (b) The revenue of the established department store is decreasing at the fastest rate. (c) The revenue of the discount store first overtakes that of the established store.

Learn how to analyze revenue trends between two department stores using derivatives and graph interpretation. This calculus problem explores key concepts like rate of change, finding where the revenue decreases the fastest or slowest, and determining when one revenue overtakes another. Suitable for advanced high school students in Grades 11-12.

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