A Function Can Have More Than One Point Where Derivative Is Zero

4. Applications of Derivatives

4.5 Derivatives and the Shape of a Graph

Learning Objectives

Explain how the sign of the first derivative affects the shape of a function's graph.
State the first derivative test for critical points.
Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function's graph.
Explain the concavity test for a function over an open interval.
Explain the relationship between a function and its first and second derivatives.
State the second derivative test for local extrema.

Earlier in this chapter we stated that if a function has a local extremum at a point then must be a critical point of However, a function is not guaranteed to have a local extremum at a critical point. For example, $f(x)={x}^{3}$ has a critical point at since $f\prime (x)=3{x}^{2}$ is zero at but does not have a local extremum at Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.

The First Derivative Test

Corollary 3 of the Mean Value Theorem showed that if the derivative of a function is positive over an interval then the function is increasing over On the other hand, if the derivative of the function is negative over an interval then the function is decreasing over as shown in the following figure.

0. In other words, f is increasing. Figure b shows a function increasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f' > 0. In other words, f is increasing. Figure c shows a function decreasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f' < 0. In other words, f is decreasing. Figure d shows a function decreasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f' < 0. In other words, f is decreasing." width="487" height="548"> **Figure 1.** Both functions are increasing over the interval At each point the derivative $f\prime (x)>0.$ Both functions are decreasing over the interval At each point the derivative $f\prime (x)<0.$

A continuous function has a local maximum at point if and only if switches from increasing to decreasing at point Similarly, has a local minimum at if and only if switches from decreasing to increasing at If is a continuous function over an interval containing and differentiable over except possibly at the only way can switch from increasing to decreasing (or vice versa) at point is if ${f}^{\prime }$ changes sign as increases through If is differentiable at the only way that ${f}^{\prime }.$ can change sign as increases through is if ${f}^{\prime }(c)=0.$ Therefore, for a function that is continuous over an interval containing and differentiable over except possibly at the only way can switch from increasing to decreasing (or vice versa) is if $f\prime (c)=0$ or ${f}^{\prime }(c)$ is undefined. Consequently, to locate local extrema for a function we look for points in the domain of such that $f\prime (c)=0$ or ${f}^{\prime }(c)$ is undefined. Recall that such points are called critical points of

Note that need not have a local extrema at a critical point. The critical points are candidates for local extrema only. In (Figure), we show that if a continuous function has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point. We show that if has a local extremum at a critical point, then the sign of ${f}^{\prime }$ switches as increases through that point.

Using (Figure), we summarize the main results regarding local extrema.

This result is known as the first derivative test.

We can summarize the first derivative test as a strategy for locating local extrema.

Now let's look at how to use this strategy to locate all local extrema for particular functions.

Using the First Derivative Test to Find Local Extrema

Use the first derivative test to find the location of all local extrema for $f(x)={x}^{3}-3{x}^{2}-9x-1.$ Use a graphing utility to confirm your results.

Use the first derivative test to locate all local extrema for $f(x)=\text{−}{x}^{3}+\frac{3}{2}{x}^{2}+18x.$

Solution

has a local minimum at -2 and a local maximum at 3.

Using the First Derivative Test

Use the first derivative test to find the location of all local extrema for $f(x)=5{x}^{1\text{/}3}-{x}^{5\text{/}3}.$ Use a graphing utility to confirm your results.

Use the first derivative test to find all local extrema for $f(x)=\sqrt[3]{x-1}.$

Concavity and Points of Inflection

We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function.

(Figure)(a) shows a function with a graph that curves upward. As increases, the slope of the tangent line increases. Thus, since the derivative increases as increases, ${f}^{\prime }$ is an increasing function. We say this function is concave up. (Figure)(b) shows a function that curves downward. As increases, the slope of the tangent line decreases. Since the derivative decreases as increases, ${f}^{\prime }$ is a decreasing function. We say this function is concave down.

In general, without having the graph of a function how can we determine its concavity? By definition, a function is concave up if ${f}^{\prime }$ is increasing. From Corollary 3, we know that if ${f}^{\prime }$ is a differentiable function, then ${f}^{\prime }$ is increasing if its derivative $f\text{″}(x)>0.$ Therefore, a function that is twice differentiable is concave up when $f\text{″}(x)>0.$ Similarly, a function is concave down if ${f}^{\prime }$ is decreasing. We know that a differentiable function ${f}^{\prime }$ is decreasing if its derivative $f\text{″}(x)<0.$ Therefore, a twice-differentiable function is concave down when $f\text{″}(x)<0.$ Applying this logic is known as the concavity test.

We conclude that we can determine the concavity of a function by looking at the second derivative of In addition, we observe that a function can switch concavity ((Figure)). However, a continuous function can switch concavity only at a point if $f\text{″}(x)=0$ or $f\text{″}(x)$ is undefined. Consequently, to determine the intervals where a function is concave up and concave down, we look for those values of where $f\text{″}(x)=0$ or $f\text{″}(x)$ is undefined. When we have determined these points, we divide the domain of into smaller intervals and determine the sign of $f\text{″}$ over each of these smaller intervals. If $f\text{″}$ changes sign as we pass through a point then changes concavity. It is important to remember that a function may not change concavity at a point even if $f\text{″}(x)=0$ or $f\text{″}(x)$ is undefined. If, however, does change concavity at a point and is continuous at we say the point is an inflection point of

Testing for Concavity

We now summarize, in (Figure), the information that the first and second derivatives of a function provide about the graph of and illustrate this information in (Figure).

What Derivatives Tell Us about Graphs
Sign of $f\prime$	Sign of $f\text{″}$	Is increasing or decreasing?	Concavity
Positive	Positive	Increasing	Concave up
Positive	Negative	Increasing	Concave down
Negative	Positive	Decreasing	Concave up
Negative	Negative	Decreasing	Concave down

The Second Derivative Test

The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.

We know that if a continuous function has a local extrema, it must occur at a critical point. However, a function need not have a local extrema at a critical point. Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. Let be a twice-differentiable function such that ${f}^{\prime }(a)=0$ and $f\text{″}$ is continuous over an open interval containing Suppose $f\text{″}(a)<0.$ Since $f\text{″}$ is continuous over $f\text{″}(x)<0$ for all $x\in I$ ((Figure)). Then, by Corollary 3, ${f}^{\prime }$ is a decreasing function over Since ${f}^{\prime }(a)=0,$ we conclude that for all $x\in I,{f}^{\prime }(x)>0$ if and ${f}^{\prime }(x)<0$ if Therefore, by the first derivative test, has a local maximum at On the other hand, suppose there exists a point such that ${f}^{\prime }(b)=0$ but $f\text{″}(b)>0.$ Since $f\text{″}$ is continuous over an open interval containing then $f\text{″}(x)>0$ for all $x\in I$ ((Figure)). Then, by Corollary $3,{f}^{\prime }$ is an increasing function over Since ${f}^{\prime }(b)=0,$ we conclude that for all $x\in I,$ ${f}^{\prime }(x)<0$ if and ${f}^{\prime }(x)>0$ if Therefore, by the first derivative test, has a local minimum at

Note that for case iii. when $f\text{″}(c)=0,$ then may have a local maximum, local minimum, or neither at For example, the functions $f(x)={x}^{3},$ $f(x)={x}^{4},$ and $f(x)=\text{−}{x}^{4}$ all have critical points at In each case, the second derivative is zero at However, the function $f(x)={x}^{4}$ has a local minimum at whereas the function $f(x)=\text{−}{x}^{4}$ has a local maximum at and the function $f(x)={x}^{3}$ does not have a local extremum at

Let's now look at how to use the second derivative test to determine whether has a local maximum or local minimum at a critical point where ${f}^{\prime }(c)=0.$

Using the Second Derivative Test

Use the second derivative to find the location of all local extrema for $f(x)={x}^{5}-5{x}^{3}.$

We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. In the next section we discuss what happens to a function as $x\to \text{±}\infty .$ At that point, we have enough tools to provide accurate graphs of a large variety of functions.

Key Concepts

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

Solution

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

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

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

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 for some value of

Solution

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

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

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

8. 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).

9. 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 touching the x axis at (−1, 0). It then increases a little before decreasing and crossing the x axis at the origin. The function then decreases to a local minimum before increasing, crossing the x-axis at (1, 0), and continuing to increase.

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.

Solution

Decreasing for increasing for

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).

For the following exercises, analyze the graphs of ${f}^{\prime },$ then list all intervals where

is increasing and decreasing and
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).

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.

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, analyze the graphs of ${f}^{\prime },$ then list all inflection points and intervals 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.

Solution

Concave up on all 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.

Solution

Concave up on all 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.

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

24. ${f}^{\prime }(x)>0$ over $x>2,-3<x<-1,{f}^{\prime }(x)<0$ over $-1<x<2,f\text{″}(x)<0$ for all

Solution

Answers will vary

26. There is a local maximum at local minimum at and the graph is neither concave up nor concave down.

Solution

Answers will vary

For the following exercises, determine

intervals where is increasing or decreasing and
local minima and maxima of

28. $f(x)= \sin x+{ \sin }^{3}x$ over $\text{−}\pi <x<\pi$

29. $f(x)={x}^{2}+ \cos x$

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

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

For the following exercises, determine

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

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

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

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

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

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

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

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

For the following exercises, determine

intervals where is increasing or decreasing,
local minima and maxima of
intervals where is concave up and concave down, and
the inflection points of 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 $x=\left[-1,1\right]$

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 $(-\frac{\pi }{2},\frac{\pi }{2})$

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

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

44. $f(x)= \sin (x){e}^{x}$ over $x=\left[\text{−}\pi ,\pi \right]$

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

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

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

For the following exercises, interpret the sentences in terms of $f,{f}^{\prime },\text{ and }f\text{″}.$

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

Solution

$f>0,{f}^{\prime }>0,f\text{″}<0$

49. A bike accelerates faster, but a car goes faster. Here Bike's position minus Car's position.

50. The airplane lands smoothly. Here is the plane's altitude.

Solution

$f>0,{f}^{\prime }<0,f\text{″}<0$

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

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

Solution

$f>0,{f}^{\prime }>0,f\text{″}>0$

For the following exercises, consider a third-degree polynomial 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. for some $1\le x\le 3$

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

Solution

True, by the Mean Value Theorem

55. There is no absolute maximum at

56. If has three roots, then it has 1 inflection point.

Solution

True, examine derivative

57. If has one inflection point, then it has three real roots.

A Function Can Have More Than One Point Where Derivative Is Zero

Source: https://opentextbc.ca/calculusv1openstax/chapter/derivatives-and-the-shape-of-a-graph/

A Function Can Have More Than One Point Where Derivative Is Zero

4.5 Derivatives and the Shape of a Graph

Learning Objectives

The First Derivative Test

Using the First Derivative Test to Find Local Extrema

Solution

Using the First Derivative Test

Concavity and Points of Inflection

Testing for Concavity

The Second Derivative Test

Using the Second Derivative Test

Key Concepts

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

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