Question 1

Suppose $$f(x) = x^3$ - 3x^2$ + 2. $$Using the First Derivative Test, at which $$x-$$value does $$f(x)$$ have a local maximum?

Accepted Answer

$$x = 1$$

Question 2

Given $$f(x) = e^x - x$$, use the Second Derivative Test to determine the nature of the critical point at $$x = 0.$$

Accepted Answer

Local minimum

Question 3

For the function $$f(x) = x^4$, what can you say about the point x = 0$?

Accepted Answer

It is a local minimum but not an inflection point

Question 4

A pharmaceutical company models the profit from a new drug as $$P(x) = -2x^3$ + 12x^2$ - 18x + 5$$, where $$x is $$the number of thousands of units produced. At what value of $$x$$ does the profit reach a local minimum?

Accepted Answer

$$x = 3$$

Question 5

Which of the following statements about the Second Derivative Test is TRUE?

Accepted Answer

If $$f'(c) = 0$$ and $$f''(c) = 0$$, the test is inconclusive

Question 6

Let $$f(x) = x^3$. What is the concavity of f(x$$)$$ at x = 0$?

Accepted Answer

No concavity; $$x = 0 is an $$inflection point

Question 7

A company’s cost function is $$C(x) = x^4$ - 4x^3$ + 6x^2$. Find all inflection points.

Accepted Answer

No inflection points

Question 8

Given $$f(x) = x e^x$$, find the $$x-$$value where the function changes concavity.

Accepted Answer

$$x = -2$$

Question 9

Suppose $$f(x) = 2\cos x. At $$which $$x in$$ $$[0, 2\pi]$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = \pi$$

Question 10

A population model is $$P(t) = t^3$ - 6t^2$ + 9t + 5. At $$what time $$t$$ does the population reach a local maximum?

Accepted Answer

$$t = 1$$

Question 11

If $$f''(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$concave up

Question 12

Given $$f(x) = 12x^3$ - 12x^2$ - 24x$$, use the Second Derivative Test to determine the nature of the critical point at $$x = 2.$$

Accepted Answer

Local minimum

Question 13

Which of the following is TRUE about inflection points?

Accepted Answer

They occur where $$f''(x) = 0$$ and the concavity changes

Question 14

A chemical reaction’s rate is modeled by $$R(t) = t^4$ - 8t^3$ + 18t^2$. Find the intervals where the reaction rate is concave up.

Accepted Answer

$$(-\infty, 1) \cup (3, \infty)$$

Question 15

If $$f(x) = x^2$ e^x$$, at which $$x-$$value does $$f(x)$$ have an inflection point?

Accepted Answer

$$x = -1$$

Question 16

Suppose $$f(x)$$ has $$f'(2) = 0$$ and $$f''(2) \u003c 0. $$What does this imply about $$f(x) at$$ $$x = 2$$?

Accepted Answer

There is a local maximum at $$x = 2$$

Question 17

A medical researcher models the spread of a disease as $$S(t) = t^3$ - 9t^2$ + 24t. At $$what time $$t$$ does the spread rate change from increasing to decreasing?

Accepted Answer

$$t = 4$$

Question 18

Given $$f(x) = x^3$ - 6x^2$ + 9x$$, find all inflection points.

Accepted Answer

$$x = 2$$

Question 19

If $$f(x) = x^5$, what is the nature of x = 0$?

Accepted Answer

Inflection point

Question 20

A company’s revenue is modeled by $$R(x) = -x^3$ + 6x^2$ - 9x + 4. At $$what value of $$x$$ does the revenue reach a local maximum?

Accepted Answer

$$x = 1$$

Second Derivative Test, Concavity, and Inflection Points

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