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

$$x  3$$

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 $$f''(x) \u003e 0$$ 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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