Question 1

Which of the following statements about the second derivative $$f''(x) is $$TRUE?

Accepted Answer

If $$f''(x) \u003e 0$$, the graph of $$f is $$concave upward at $$x.$$

Question 2

Suppose $$y = s(t)$$ describes the position of a car along a straight road. If $$v(t) = s'(t) \u003e 0$$ and $$a(t) = s''(t) \u003c 0$$, what is happening to the car?

Accepted Answer

The car is moving forward and slowing down.

Question 3

Given $$f(x) = x^4$ - 4x^3$, find all inflection points.

Accepted Answer

At $$x = 0$$ and $$x = 2$$

Question 4

If $$f(x) is a $$polynomial of degree $$m$$, what is the degree of its $$nth $$derivative $$f^{(n)}(x$$)$$?

Accepted Answer

It is $$m-n$$

Question 5

For the function $$f(x) = e$$^{2x}$$, what is f$$^{(100)}$$(x)$$?

Accepted Answer

$$2^{100}$$ $$e^{2x}$$

Question 6

A point $$P = (c, f(c)) is an $$inflection point of $$f if$$:

Accepted Answer

$$f''(c)$$ changes sign at $$x = c$$

Question 7

Suppose $$f''(x) = 0 at$$ $$x = c$$ and $$f''(x)$$ changes from positive to negative as $$x$$ increases through $$c. $$What does this indicate about $$f at$$ $$x = c$$?

Accepted Answer

There is an inflection point at $$x = c$$

Question 8

If $$v(t)  0$$ for an object moving along a line, what is happening?

Accepted Answer

The object is moving in the negative direction and slowing down.

Question 9

Which of the following is the correct notation for the second derivative of $$f(x)$$?

Accepted Answer

$$\frac{d^2$}{dx^2$} f(x)$$

Question 10

If $$f(x) is a $$linear function, what is $$f''(x)$$?

Accepted Answer

$$f''(x) = 0$$

Question 11

For $$f(x) = x$$^{-1}$$, what is f$$^{(100)}$$(x)$$?

Accepted Answer

$$100! x$$^{-101}$$

Question 12

If $$f''(x) \u003c 0$$ for all $$x in an $$interval $$I$$, what can you conclude about the graph of $$f on$$ $$I$$?

Accepted Answer

It is concave downward on $$I$$

Question 13

Suppose $$f''(3) \u003e 0$$ and $$f'(3) = 0. $$What does the second derivative test tell you about $$f at$$ $$x = 3$$?

Accepted Answer

$$f$$ has a local minimum at $$x = 3$$

Question 14

In economics, the point of diminishing returns for a revenue function $$f(x) is $$associated with:

Accepted Answer

An inflection point of $$f(x)$$

Question 15

Which of the following is NOT a correct notation for the second derivative?

Accepted Answer

$$f^{(1)}(x$$)$$

Question 16

If $$f''(c) = 0$$ but $$f''(x)$$ does not change sign at $$x = c$$, what can you conclude?

Accepted Answer

There is neither a local extremum nor an inflection point at $$x = c$$

Question 17

For $$f(x) = e^x + x^4$ + x^3$ - x^2$ + x + 1$$, what is $$f^{(100)}(x$$)$$?

Accepted Answer

$$e^x$$

Question 18

Which of the following best describes the relationship between the sign of $$f''(x)$$ and the concavity of $$f$$?

Accepted Answer

If $$f''(x) \u003e 0$$, $$f is $$concave upward.

Question 19

If $$v(t) \u003e 0$$ and $$a(t) \u003e 0$$ for an object moving along a line, what is happening?

Accepted Answer

The object is moving in the positive direction and speeding up.

(L15) Concavity, Second Derivative, and Applications in Business Calculus

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