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

The car is moving forward and slowing down.

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

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

Q: 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 $x = c$?

There is an inflection point at $x = c$

Question 1

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

Accepted Answer

If $f''(x) > 0$, 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 $v(t) = s'(t) > 0$ and $a(t) = s''(t) < 0$, 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 $n$th 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 $v(t) < 0$ and $a(t) > 0$ 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 $f''(x) < 0$ 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 $f''(3) > 0$ 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 $f''(x) > 0$, $f$ is concave upward.

Question 19

If $v(t) > 0$ and $a(t) > 0$ 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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