he revenue (in thousands of dollars) from producing units of an item is
nodeled by .
b. Find the marginal revenue at .

Alright, I need to find the marginal revenue at for the revenue function . Let’s break this down step by step.

First, I recall that marginal revenue is the derivative of the revenue function with respect to the number of units produced, . So, I need to find .

Given:

To find the derivative, I’ll differentiate each term separately.

So, the marginal revenue function is:

Now, I need to find .

Plugging in :

Wait a second, a negative marginal revenue doesn’t make sense in this context. Let me double-check my calculations.

Reviewing the derivative:

Calculating again:

Hmm, the math checks out. The negative value suggests that at units, producing an additional unit would decrease revenue by $6,000. This could imply that the revenue function has reached its maximum at this point, and beyond this point, producing more units would start to reduce total revenue.

But let’s consider the context. If the revenue is decreasing, perhaps the company has reached the point of diminishing returns. Maybe the cost of producing additional units is higher than the revenue generated, leading to a decrease in overall profit.

Alternatively, there might be an error in the revenue function. If the revenue is supposed to increase with more production, perhaps the coefficient of should be negative, making the function concave down. But as per the given function, it’s currently concave up, which would mean that revenue increases initially but then starts to decrease after a certain point.

In conclusion, based on the provided revenue function , the marginal revenue at is $-6,000. This indicates that producing an additional unit at this production level would result in a loss of $6,000 in revenue.