Question 1

Suppose the profit function of a firm is represented by a curve on a graph. At which point on the curve does the firm maximize its profit?

Accepted Answer

At the maximum point, where the slope of the tangent is zero.

Question 2

Given a cost function $$C(q)$$, how would you mathematically determine the output level $$q$$ that minimizes cost?

Accepted Answer

Find $$q$$ such that $$\frac{dC}{dq} = 0$$ and $$\frac{d^2$C}{dq^2$} \u003e 0.$$

Question 3

If a firm's revenue function $$R(q)$$ has a maximum at $$q = 5$$, what does this imply about the slope of the revenue function at $$q = 5$$?

Accepted Answer

The slope is zero: $$\frac{dR}{dq} = 0 at$$ $$q = 5.$$

Question 4

Which of the following best describes the minimum point on a cost curve?

Accepted Answer

It is the point where the cost is lowest and the slope of the tangent is zero.

Question 5

A student claims that the maximum point on a graph always occurs at the highest value of $$x. Is $$this statement correct?

Accepted Answer

No, the maximum point is where the function reaches its highest value, not necessarily at the largest $$x.$$

Question 6

If a function $$f(x)$$ has a minimum at $$x = 2$$, which of the following must be true?

Accepted Answer

$$\frac{df}{dx} = 0 at$$ $$x = 2$$ and $$\frac{d^2$f}{dx^2$} \u003e 0 at$$ $$x = 2.$$

Question 7

Which of the following is a necessary condition for a point to be a maximum or minimum on a graph?

Accepted Answer

The first derivative at that point must be zero.

Question 8

A firm’s profit function is $$\pi(q) = -q^2$ + 6q. At $$what value of $$q is $$profit maximized?

Accepted Answer

$$q = 3$$

Question 9

If the graph of a function opens upwards and has a single turning point, what can you say about this turning point?

Accepted Answer

It is a minimum point.

Question 10

Suppose a cost function $$C(q)$$ has a minimum at $$q = 4. $$Which of the following statements is true?

Accepted Answer

$$\frac{dC}{dq} = 0 at$$ $$q = 4$$ and $$\frac{d^2$C}{dq^2$} \u003e 0 at$$ $$q = 4.$$

Question 11

Which of the following best describes the process of finding the maximum or minimum point on a graph?

Accepted Answer

Set the first derivative equal to zero and check the second derivative.

Question 12

If a function $$f(x)$$ has $$\frac{df}{dx} = 0 at$$ $$x = 1$$ and $$\frac{d^2$f}{dx^2$} \u003c 0 at$$ $$x = 1$$, what does this indicate?

Accepted Answer

There is a maximum at $$x = 1.$$

Question 13

A real-world example: A company wants to minimize its transportation costs. Which mathematical approach should it use?

Accepted Answer

Find the minimum point of the cost function by setting the first derivative to zero and confirming the second derivative is positive.

Question 14

If a function $$f(x)$$ has a maximum at $$x = 2$$ and a minimum at $$x = 5$$, which of the following is true?

Accepted Answer

$$\frac{df}{dx} = 0 at $$both $$x = 2$$ and $$x = 5.$$

Question 15

Which of the following is NOT a characteristic of a minimum point on a graph?

Accepted Answer

The slope of the tangent is positive.

Finding Maximum and Minimum Points on Graphs

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