Question 1

Which of the following best describes the concavity of the function shown in the first image?

Accepted Answer

The function is concave up for all $$x.$$

Question 2

For the function shown in the second image, what can you conclude about the second derivative $$f''(x)$$?

Accepted Answer

$$f''(x) \u003c 0$$ for all $$x$$

Question 3

Given $$f(x) = x^4$ + 3x^3$ - 9x^2$ - 11x + 4$$, what is the second derivative $$f''(x)$$?

Accepted Answer

$$f''(x) = 12x^2$ + 18x - 18$$

Question 4

Suppose $$f'(c) = 0$$ and $$f''(c) \u003e 0. $$According to the Second Derivative Test, what can you conclude about $$f at$$ $$x = c$$?

Accepted Answer

$$f$$ has a relative minimum at $$x = c$$

Question 5

If $$f''(x)$$ changes sign at $$x = a$$, what does this indicate about the graph of $$f$$?

Accepted Answer

$$x = a is a $$candidate for an inflection point

Question 6

Given $$s(t) = -16t^2$ + 905t$$, what is the acceleration of the object at any time $$t$$?

Accepted Answer

$$a(t) = -32$$

Question 7

For the function $$f(x) = x^3$ - 3x^2$ + 5$$, find all inflection points.

Accepted Answer

There is an inflection point at $$x = 1$$

Question 8

Which of the following statements is true about the velocity and acceleration of an object moving away from its starting point?

Accepted Answer

Velocity is positive and acceleration is positive if the object is speeding up

Question 9

If $$f''(x) \u003e 0 on an $$interval, what does this tell you about the graph of $$f on $$that interval?

Accepted Answer

The graph is concave up

Question 10

Given $$f(x) = (x-4)^2$, use the Second Derivative Test to classify the critical point.

Accepted Answer

Relative minimum at $$x = 4$$

Question 11

Suppose $$f''(x) = 0 at$$ $$x = 2$$, but $$f''(x)$$ does not change sign at $$x = 2. $$What can you conclude?

Accepted Answer

$$x = 2 is $$not an inflection point

Question 12

If $$f(x) is $$concave down on $$(-\infty, \infty)$$, which of the following could be its graph?

Accepted Answer

A downward-opening parabola

Question 13

Find the third derivative of $$g(t) = e$$^{3t}$$.

Accepted Answer

$$g'''(t) = 27e$$^{3t}$$

Question 14

A company models its profit with $$P(x) = -2x^2$ + 8x + 10. At $$what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 2$$

Question 15

Given $$y = (5x + 1)^2$, what is the second derivative y$$''$$?

Accepted Answer

$$y'' = 50$$

Higher Derivatives, Concavity, and the Second Derivative Test

Study Guide - Practice Questions