Business Calculus Quiz: Concavity And Inflection Points

A company models its profit, $P(x)$, in thousands of dollars, from spending $x$ thousand dollars on advertising as $P(x) = -x^3 + 21x^2 + 90x + 500$ for $x \ge 0$. The company wishes to identify the point of diminishing returns, which is the advertising expenditure at which the rate of growth of profit begins to decrease. What is this expenditure level?

The total cost to produce $q$ units of a specialized electronic component is given by the function $C(q) = 0.05q^3 - 4.5q^2 + 150q + 1000$. At what production level is the marginal cost, $C'(q)$, at a minimum?

Let $P(t)$ represent the number of units a new employee produces on day $t$ of their employment. An efficiency expert observes that for a particular employee on day $t_0=10$, $P'(10) > 0$ and $P''(10) < 0$. Which of the following is the best interpretation of this observation?

The total number of subscribers to a streaming service, $N(t)$, in millions, is modeled by the function $N(t) = -0.5t^3 + 9t^2 + 20t$ where $t$ is the number of years since the service launched, for $0 \le t \le 15$. At what time is the subscriber base growing the fastest?

The marginal profit for a product is found to be increasing for production levels between 0 and 1,000 units. For production levels above 1,000 units, the marginal profit is positive but decreasing. Which statement provides the most accurate description of the total profit function, $P(q)$, at the production level $q=1,000$?

A manufacturing company's profit function is $P(x) = -0.5x^3 + 15x^2 - 120x + 500$, where $x$ represents the number of units produced (in thousands) and $P(x)$ is profit in thousands of dollars. The company's financial analyst needs to determine the production level where the rate of profit increase begins to slow down most significantly. At what production level does this occur?

Let $C(x)$ be a differentiable total cost function for producing $x$ units of a product, with $x>0$. If the average cost function, $AC(x) = C(x)/x$, reaches its minimum value at a production level $x_0$, which of the following statements about the total cost function $C(x)$ at $x_0$ must be true?

A financial analyst determines that for a specific investment of $x$ dollars, the profit function $P(x)$ has the properties $P''(5000) = 0$ and $P'''(5000) = -2$. Based on the third derivative test, what is the most definitive conclusion about the profit function at an investment level of x = \5,000?

Two competing advertising campaigns, A and B, generate profit over time $t$ (in weeks) according to the functions $P_A(t) = -2t^3 + 30t^2$ and $P_B(t) = -t^3 + 24t^2$. For what initial time interval is the profit from campaign A accelerating faster than the profit from campaign B?

A firm's total revenue, $R(x)$, from selling $x$ thousand units is given by $R(x) = -0.02x^3 + 3.6x^2 + 50x$ for $0 \le x \le 150$. Over which production interval is the firm experiencing accelerating growth in revenue?

The total cost for a warehouse to store $x$ tons of a commodity is given by $C(x) = 0.0002x^3 - 0.24x^2 + 120x + 5000$ for $0 \le x \le 600$. Due to capacity constraints and handling difficulties, the marginal storage cost eventually begins to increase. At what storage level $x$ does this occur?

A fintech company's user acquisition cost varies with marketing intensity according to $U(m) = m^4 - 8m^3 + 18m^2 - 8m + 50$, where $m$ represents marketing intensity (scale 0-5) and $U(m)$ is cost per user in dollars. The growth team needs to identify all points where the rate of change in acquisition cost transitions between accelerating and decelerating behaviors. How many such transition points exist in the domain $0 \leq m \leq 5$?

An e-commerce company's customer acquisition cost as a function of marketing spend is $A(s) = \frac{500s}{s^2 + 100} + 50$, where $s$ is marketing spend in thousands of dollars and $A(s)$ is the cost per customer in dollars. The marketing team wants to identify the spend level where the rate of change of acquisition cost transitions from concave behavior to convex behavior. What characterizes this transition point?