Question 1

Suppose the cost function for producing $x$ units of a product is given by $C(x) = 5x^3 - 30x^2 + 60x + 100$. What is the second derivative $C''(x)$, and what does it represent in a business context?

Accepted Answer

$C''(x) = 30x - 60$, representing the rate of change of marginal cost

Question 2

A company’s revenue function is $R(x) = 100x - 2x^2$. At what value of $x$ does the revenue function change from concave up to concave down?

Accepted Answer

The revenue function is always concave down for all $x$

Question 3

Given the profit function $P(x) = -x^3 + 6x^2 + 15x + 10$, find all inflection points.

Accepted Answer

At $x = 1$ and $x = 5$

Question 4

If the demand function for a product is $p(x) = 50 - 0.5x$, what is the concavity of the total revenue function $TR(x) = p(x) \cdot x$?

Accepted Answer

The total revenue function is concave down for all $x$

Question 5

A pharmaceutical company models the cost to produce $x$ batches of a drug as $C(x) = 2000 + 100x + 0.5x^2$. What does the sign of $C''(x)$ indicate about the cost structure?

Accepted Answer

Costs are increasing at an increasing rate (diseconomies of scale)

Question 6

Given the function $f(x) = x^4 - 4x^3 + 6x^2$, determine the intervals where $f(x)$ is concave up.

Accepted Answer

$x < 1$ and $x > 3$

Question 7

A company’s profit function is $P(x) = 2x^3 - 9x^2 + 12x + 5$. At what value(s) of $x$ does the profit function have an inflection point?

Accepted Answer

At $x = 1.5$

Question 8

Suppose the marginal cost function for a product is $MC(x) = 12x^2 - 24x + 10$. What is the rate of change of the marginal cost when $x = 2$?

Accepted Answer

$MC'(2) = 24$

Question 9

A business models its revenue as $R(x) = 80x - x^2$. What is the maximum revenue, and at what value of $x$ does it occur?

Accepted Answer

Maximum revenue is $1600$ at $x = 40$

Question 10

If the second derivative of a cost function $C''(x)$ is negative for all $x$, what does this imply about the cost function?

Accepted Answer

The cost function is concave down everywhere

Question 11

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the marginal profit when $x = 10$?

Accepted Answer

$P'(10) = 0$

Question 12

Given the function $f(x) = 3x^3 - 12x^2 + 12x + 5$, find the $x$-values where the function changes concavity.

Accepted Answer

At $x = 2$

Question 13

A manufacturer’s cost function is $C(x) = 100 + 20x + x^2$. What is the marginal cost when $x = 5$?

Accepted Answer

$C'(5) = 30$

Question 14

If a function $f(x)$ has $f''(x) > 0$ for all $x$ in its domain, which of the following statements is true?

Accepted Answer

$f(x)$ is concave up everywhere

Question 15

A company’s revenue function is $R(x) = 60x - 0.5x^2$. What is the marginal revenue when $x = 20$?

Accepted Answer

$R'(20) = 50$

Question 16

Suppose the profit function is $P(x) = -x^3 + 9x^2 - 24x + 20$. At what value(s) of $x$ does the function have an inflection point?

Accepted Answer

At $x = 3$

Question 17

A business’s cost function is $C(x) = 500 + 10x + 0.2x^2$. What is the marginal cost at $x = 50$?

Accepted Answer

$C'(50) = 20$

Question 18

If the second derivative of a profit function is positive at a certain point, what does this indicate about the graph of the profit function at that point?

Accepted Answer

The graph is concave up at that point

Question 19

A company’s revenue function is $R(x) = 120x - x^2$. What is the value of $x$ that maximizes revenue, and what is the concavity at that point?

Accepted Answer

$x = 60$, and the function is concave down at that point

Question 20

Given the cost function $C(x) = 2x^3 - 15x^2 + 36x + 100$, at what value(s) of $x$ does the cost function have an inflection point?

Accepted Answer

At $x = 2.5$

Higher Derivatives and Concavity (Business Calculus Study Notes)

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