Problem Set: Derivatives and the Shape of a Graph

1. If $c$ is a critical point of $f(x)$ , when is there no local maximum or minimum at $c$ ? Explain.

2. For the function $y=x^3$ , is $x=0$ both an inflection point and a local maximum/minimum?

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It is not a local maximum/minimum because $f^{\prime}$ does not change sign

3. For the function $y=x^3$ , is $x=0$ an inflection point?

4. Is it possible for a point $c$ to be both an inflection point and a local extrema of a twice differentiable function?

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5. Why do you need continuity for the first derivative test? Come up with an example.

6. Explain whether a concave-down function has to cross $y=0$ for some value of $x$ .

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False; for example, $y=\sqrt{x}$ .

7. Explain whether a polynomial of degree 2 can have an inflection point.

For the following exercises (8-12), analyze the graphs of $f^{\prime}$ , then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−2, 0). Then it continues increasing a little before decreasing and crossing the x axis at (−1, 0). It achieves a local minimum at (1, −6) before increasing and crossing the x axis at (2, 0).

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Increasing for [latex]-22[/latex]; decreasing for

x < - 2

$x<-2$ and [latex]-1

10.

The function f’(x) is graphed. The function starts negative and touches the x axis at the origin. Then it decreases a little before increasing to cross the x axis at (1, 0) and continuing to increase.

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Decreasing for

x < 1

$x<1$ ; increasing for

x > 1

$x>1$

11.

The function f’(x) is graphed. The function starts positive and decreases to touch the x axis at (−1, 0). Then it increases to (0, 4.5) before decreasing to touch the x axis at (1, 0). Then the function increases.

12.

The function f’(x) is graphed. The function starts at (−2, 0), decreases to (−1.5, −1.5), increases to (−1, 0), and continues increasing before decreasing to the origin. Then the other side is symmetric: that is, the function increases and then decreases to pass through (1, 0). It continues decreasing to (1.5, −1.5), and then increase to (2, 0).

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For the following exercises (13-17), analyze the graphs of $f^{\prime}$ , then list

all intervals where $f$ is increasing and decreasing and
where the minima and maxima are located.

13.

The function f’(x) is graphed. The function starts at (−2, 0), decreases for a little and then increases to (−1, 0), continues increasing before decreasing to the origin, at which point it increases.

14.

The function f’(x) is graphed. The function starts at (−2, 0), increases and then decreases to (−1, 0), decreases and then increases to an inflection point at the origin. Then the function increases and decreases to cross (1, 0). It continues decreasing and then increases to (2, 0).

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a. Increasing over [latex]-22[/latex]; decreasing over [latex]x<-2 \, -1

b. maxima at $x=-1$ and $x=1$ , minima at $x=-2$ and $x=0$ and $x=2$

15.

The function f’(x) is graphed from x = −2 to x = 2. It starts near zero at x = −2, but then increases rapidly and remains positive for the entire length of the graph.

16.

The function f’(x) is graphed. The function starts negative and crosses the x axis at the origin, which is an inflection point. Then it continues increasing.

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a. Increasing over

x > 0

$x>0$ , decreasing over

x < 0

$x<0$ ;b. Minimum at

x = 0

$x=0$

17.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing a little before decreasing and touching the x axis at the origin. It increases again and then decreases to (1, 0). Then it increases.

For the following exercises (18-22), analyze the graphs of $f^{\prime}$ , then list all inflection points and intervals $f$ that are concave up and concave down.

18.

The function f’(x) is graphed. The function is linear and starts negative. It crosses the x axis at the origin.

Show Solution

Concave up on all

x

$x$ , no inflection points

19.

The function f’(x) is graphed. It is an upward-facing parabola with 0 as its local minimum.

20.

The function f’(x) is graphed. The function resembles the graph of x3: that is, it starts negative and crosses the x axis at the origin. Then it continues increasing.

Show Solution

Concave up on all

x

$x$ , no inflection points

21.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−0.5, 0). Then it continues increasing to (0, 1.5) before decreasing and touching the x axis at (1, 0). It then increases.

22.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing to a local maximum at (0, 1), at which point it decreases and touches the x axis at (1, 0). It then increases.

Show Solution

Concave up for

x < 0

$x<0$ and

x > 1

$x>1$ , concave down for [latex]0

For the following exercises (23-27), draw a graph that satisfies the given specifications for the domain $x=[-3,3]$ . The function does not have to be continuous or differentiable.

23. $f(x)>0, \, f^{\prime}(x)>0$ over [latex]x>1, \, -3

24. $f^{\prime}(x)>0$ over [latex]x>2, \, -3

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25. $f^{\prime \prime}(x)<0$ over [latex]-10, \, -3

26. There is a local maximum at $x=2$ , local minimum at $x=1$ , and the graph is neither concave up nor concave down.

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27. There are local maxima at $x=\pm 1$ , the function is concave up for all $x$ , and the function remains positive for all $x$ .

For the following exercise, determine a. intervals where $f$ is concave up or concave down, and b. the inflection points of $f$ .

30. $f(x)=x^3-4x^2+x+2$

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a. Concave up for $x>\frac{4}{3}$ , concave down for $x<\frac{4}{3}$ b. Inflection point at $x=\frac{4}{3}$

For the following exercises (31-37), determine

intervals where $f$ is increasing or decreasing,
local minima and maxima of $f$ ,
intervals where $f$ is concave up and concave down, and
the inflection points of $f$ .

31. $f(x)=x^2-6x$

32. $f(x)=x^3-6x^2$

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a. Increasing over $x<0$ and $x>4$ , decreasing over [latex]02[/latex], concave down for $x<2$ d. Infection point at $x=2$

33. $f(x)=x^4-6x^3$

34. $f(x)=x^{11}-6x^{10}$

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a. Increasing over $x<0$ and $x>\frac{60}{11}$ , decreasing over [latex]0\frac{54}{11}[/latex] d. Inflection point at $x=\frac{54}{11}$

35. $f(x)=x+x^2-x^3$

36. $f(x)=x^2+x+1$

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a. Increasing over $x>-\frac{1}{2}$ , decreasing over $x<-\frac{1}{2}$ b. Minimum at $x=-\frac{1}{2}$ c. Concave up for all $x$ d. No inflection points

37. $f(x)=x^3+x^4$

For the following exercises (38-47), determine

intervals where $f$ is increasing or decreasing,
local minima and maxima of $f$ ,
intervals where $f$ is concave up and concave down, and
the inflection points of $f$ . Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

38. [T] $f(x)= \sin (\pi x)- \cos (\pi x)$ over $x=[-1,1]$

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a. Increases over [latex]-\frac{1}{4}\frac{3}{4}[/latex] and $x<-\frac{1}{4}$ b. Minimum at $x=-\frac{1}{4}$ , maximum at $x=\frac{3}{4}$ c. Concave up for [latex]-\frac{3}{4}\frac{1}{4}[/latex] d. Inflection points at $x=-\frac{3}{4}, \, x=\frac{1}{4}$

39. [T] $f(x)=x + \sin (2x)$ over $x=\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

40. [T] $f(x)= \sin x+ \tan x$ over $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

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a. Increasing for all $x$ b. No local minimum or maximum c. Concave up for $x>0$ , concave down for $x<0$ d. Inflection point at $x=0$

41. [T] $f(x)=(x-2)^2 (x-4)^2$

42. [T] $f(x)=\dfrac{1}{1-x}, \, x \ne 1$

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a. Increasing for all $x$ where defined b. No local minima or maxima c. Concave up for $x<1$ , concave down for $x>1$ d. No inflection points in domain

43. [T] $f(x)=\dfrac{\sin x}{x}$ over $x=[-2\pi ,0) \cup (0,2\pi]$

44. $f(x)= \sin x e^x$ over $x=[−\pi ,\pi]$

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a. Increasing over [latex]-\frac{\pi }{4}\frac{3\pi }{4}, \, x<-\frac{\pi }{4}[/latex] b. Minimum at $x=-\frac{\pi }{4}$ , maximum at $x=\frac{3\pi }{4}$ c. Concave up for [latex]-\frac{\pi }{2}\frac{\pi }{2}[/latex] d. Infection points at $x=\pm \frac{\pi }{2}$

45. $f(x)=\ln x \sqrt{x}, \, x>0$

46. $f(x)=\frac{1}{4}\sqrt{x}+\frac{1}{x}, \, x>0$

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a. Increasing over $x>4$ , decreasing over [latex]08\sqrt[3]{2}[/latex] d. Inflection point at $x=8\sqrt[3]{2}$

47. $f(x)=\dfrac{e^x}{x}, \, x\ne 0$

For the following exercises (48-52), interpret the sentences in terms of $f, \, f^{\prime}$ , and $f^{\prime \prime}$ .

48. The population is growing more slowly. Here $f$ is the population.

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$f>0, \, f^{\prime}>0, \, f^{\prime \prime}<0$

49. A bike accelerates faster, but a car goes faster. Here $f$ represents the Bike’s position minus the Car’s position.

50. The airplane lands smoothly. Here $f$ is the plane’s altitude.

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$f>0, \, f^{\prime}<0, \, f^{\prime \prime}<0$

51. Stock prices are at their peak. Here $f$ is the stock price.

52. The economy is picking up speed. Here $f$ is a measure of the economy, such as GDP.

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$f>0, \, f^{\prime}>0, \, f^{\prime \prime}>0$

For the following exercises (53-57), consider a third-degree polynomial $f(x)$ , which has the properties $f^{\prime}(1)=0, \, f^{\prime}(3)=0$ . Determine whether the following statements are true or false. Justify your answer.

53. $f(x)=0$ for some $1 \le x \le 3$

54. $f^{\prime \prime}(x)=0$ for some $1 \le x \le 3$

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55. There is no absolute maximum at $x=3$

56. If $f(x)$ has three roots, then it has 1 inflection point.

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57. If $f(x)$ has one inflection point, then it has three real roots.

Applications of Derivatives: Supplemental Content

Problem Set: Derivatives and the Shape of a Graph

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