Arrange the following events in the correct chronological and causal order to illustrate the shift in the textile trade between India and Britain from the early 18th to the mid-19th century.

A firm's profit is a function of its production level, q. An analyst has identified a production level of q = 20 units as a point where profit is potentially optimized (i.e., the first derivative is zero). The second derivative of the profit function is given by the formula d²P/dq² = 100 - 6q. Based on this information, what can be concluded about the profit at q = 20?

Optimizing Production Output

A marketing analyst is trying to determine the advertising budget (A, in thousands of dollars) that maximizes a company's revenue. They identify a potential maximum at A = 30. The second derivative of the revenue function with respect to the advertising budget is given by d²R/dA² = 150 - 6A. The analyst concludes that A = 30 is indeed a point of maximum revenue.

Confirming Optimal Fertilizer Use

An economist is analyzing a function to find its optimal points. They have already found the points where the first derivative is zero. Match the result of the second derivative test at one of these points with the correct conclusion about the function's value at that point.

An analyst is trying to find the production level that maximizes a company's profit. After finding a production level where the first derivative of the profit function is zero, they calculate the second derivative at that same point and get a value of -450. Because this value is ______, the analyst can confirm that the production level corresponds to a local profit maximum.

The Role of the Second Derivative in Confirming Optimal Choices

An economics student is analyzing a model to find the number of hours (h) per day that maximizes a person's well-being. They've identified h = 8 as a potential optimum. To verify if it's a maximum, they use the second derivative test. The second derivative of the well-being function is given by d²W/dh² = 20(15 - 2h).

The student's analysis is as follows:

1. Substitute h = 8 into the formula: 20(15 - 2 * 8).
2. Calculate the result: 20(15 - 16) = 20(-1) = -20.
3. Conclusion: Since the result is a negative number, h = 8 corresponds to a point of minimum well-being.

Which part of the student's analysis is flawed?

Evaluating a Pricing Strategy

A marketing analyst is trying to determine the advertising budget (A, in thousands of dollars) that maximizes a company's revenue. They identify a potential maximum at A = 30. The second derivative of the revenue function with respect to the advertising budget is given by d²R/dA² = 150 - 6A. The analyst concludes that A = 30 is indeed a point of maximum revenue.