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Chapter 3: Applications of Differentiation
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Section 3.7: What Derivatives Reveal about Graphs
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Essentials



Introduction


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•

Figure 3.7.1, an image of the
tutor, illustrates the features of the graph of $f\left(x\right)\=x\mathrm{cos}\left(x\right)$ that can be determined from $f$ itself, and from the derivatives $f\prime$ and $f\u2033$.

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Where $f$ is increasing or decreasing, its graph is drawn in red or black, respectively,

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Intervals where the graph of $f$ is concave up or down are shaded in gray or yellow, respectively.

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Relative extrema and inflection points are shown in green.

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Selecting one of the eight radiobuttons and clicking the "Calculate" button yields the information listed in Table 3.7.1.

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Figure 3.7.2 uses the FunctionChart (a.k.a. FunctionPlot) command to draw the figure shown in Figure 3.7.1. The command provides slightly more control over the features of the graph. The symbols for the seven green points can be made larger, and arrows are used to indicate concavity.

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The $x$intercepts are marked with circles; the inflection points, with crosses; and the extreme points with diamonds. These distinctions are not visible in the tutor.

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Figure 3.7.1 Curve Analysis tutor applied to $x\mathrm{cos}\left(x\right)$




>

Student:SetColors(red,black,green,gray,yellow):
Student:Calculus1:FunctionChart(x*cos(x),x=0..2*Pi,pointoptions=[symbolsize=20],caption=[],concavity=[filled(gray,yellow)]);


Figure 3.7.2 Graph via the FunctionChart command






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The default color scheme has been modified to improve visibility in both figures and in the tutor itself.

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The terms increasing, decreasing, extrema, relative extrema, concavity, and inflection point are defined below.

Table 3.7.1 displays the information that would be provided by the "Calculate" button in the tutor.
The local maxima occur at:
[.860, .561]
[6.28, 6.28]

The local minima occur at:
[0., 0.]
[3.43, 3.29]

The function is increasing on the intervals:
[0., .860]
[3.43, 6.28]

The function is decreasing on the interval:
[.860, 3.43]

The function is concave up on the interval:
[2.29, 5.09]

The function is concave down on the intervals:
[0., 2.29]
[5.09, 6.28]

The points of inflection occur at:
[2.29, 1.51]
[5.09, 1.88]

The zeros occur at $x\=$:
0.
1.57
4.71

Table 3.7.1 Data generated by the Curve Analysis tutor for $f\left(x\right)\=x\mathrm{cos}\left(x\right)comma;x\in \left[0comma;2\mathrm{pi;}\right]$



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Increasing/Decreasing on an Interval


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The Curve Analysis tutor introduces the notion of "intervals of increase and decrease." Roughly speaking, if in an interval, the graph of a function $f\left(x\right)$ rises as $x$ increases, then the function increases on that interval. Alternatively, if the graph falls, then the function decreases on that interval. Definition 3.7.1 makes this precise.
Definition 3.7.1: Increasing and Decreasing on an Interval

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A function $f$ is increasing on an interval $I$ if $f\left(a\right)<f\left(b\right)$ whenever $a<b$ are in $I$.

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A function $f$ is decreasing on an interval $I$ if $f\left(a\right)\>f\left(b\right)$ whenever $a<b$ are in $I$.

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If either $a$ or $b$ is an endpoint, then $I$ is taken as halfopen and halfclosed.




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Essentially, for a function increasing on an interval, $f\left(a\right)$ is greater than $f\left(b\right)$ whenever $b$ is to the right of $a$. However, some texts define the notion of increasing "at a point." Here, if $f\left(a\right)$ is greater than all function values in some neighborhood of $a$, then the function is increasing at $a$. There are functions that increase "at a point" but do not increase in any interval containing that point. (The function $f\left(x\right)\=x\/2\+{x}^{2}\mathrm{sin}\left(1sol;x\right)$, $x\ne 0\,f\left(0\right)equals;0$, is increasing at $x\=0$, but is not an increasing function in any neighborhood of $x\=0$. It has infinite oscillations near $x\=0$, but bottoms of the wiggles rise higher and higher as $x$ increases.)
Thus, "increasing in an interval" does not mean "increasing at every point in the interval." The concepts are different and unrelated. To avoid confusion, the notion of increase or decrease at a point will not be considered further.
There is a useful connection between the first derivative of a function and the intervals where it increases and decreases. This connection is stated formally in Table 3.7.2.
First Derivative Test for Increase/Decrease of a Differentiable Function

If $f\prime \left(x\right)$ exists and is positive throughout an interval $I$, then $f$ is increasing on $I$.
If $f\prime \left(x\right)$ exists and is negative throughout an interval $I$, the $f$ is decreasing on $I$.

Table 3.7.2 Connection between the first derivative and intervals of increase or decrease



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Extrema, both Absolute and Relative


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The extrema for a function are its maximum and minimum values. A relative (a.k.a. local) extremum is a value that is extreme (maximum or minimum) relative to nearby values. An absolute (a.k.a. global) extremum is the highest or lowest value attained by a function over some domain. Extrema at a closed endpoint of an interval are considered to be relative. Absolute extrema are deduced from the set of all relative extrema. Definition 3.7.2 makes these ideas precise.
Definition 3.7.2: Absolute and Relative Extrema

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The function $f$ has a relative maximum at $x\=c$ if $f\left(c\right)\ge f\left(x\right)$ for $x$ in a neighborhood of $c$.

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The function $f$ has an absolute maximum at $x\=c$ if $f\left(c\right)\ge f\left(x\right)$ for all $x$ in the domain of $f$.

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The function $f$ has a relative minimum at $x\=c$ if $f\left(c\right)\le f\left(x\right)$ for $x$ in a neighborhood of $c$.

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The function $f$ has an absolute minimum at $x\=c$ if $f\left(c\right)\le f\left(x\right)$ for all $x$ in the domain of $f$.




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In a word, a relative (or local) extreme is the highest or lowest value a function takes in a neighborhood, whereas the absolute (or global) extreme is the highest or lowest value a function takes over its whole domain.
There are connections between relative extrema and the first and second derivatives of a function. Theorem 3.7.1 is the formal statement of the conditions under which, at a relative extremum, the first derivative vanishes. In essence, at the top of a hill or the bottom of a valley, the tangent line is horizontal, that is, it has zero slope. Boiled down to the terse "if extreme, then $f\prime \=0$," it's converse would be: "if $f\prime \left(c\right)\=0$, then $f\left(c\right)$ is extreme," which just happens to be false! There are functions for which $f\prime \left(c\right)\=0$ but $f\left(c\right)$ is not extreme!
Theorem 3.7.1 (Fermat's Theorem)

1.

$x\=c$ is an interior point of a closed interval $I$

2.

$f\left(c\right)$ is a relative extremum on $I$

3.

$f\prime \left(c\right)$ exists

⇒
1.

$f\prime \left(c\right)\=0$




A consequence of Theorem 3.7.1 is that if $f\prime \left(c\right)\=0$ or $f\prime \left(c\right)$ does not exist, then $f\left(c\right)$ should be inspected as a candidate for an extreme value. This leads to the definition of a critical number.
Definition 3.7.3: Critical Number

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If $x\=c$ is in the domain of $f$ and either $f\prime \left(c\right)\=0$ or $f\prime \left(c\right)$ does not exist, $x\=c$ is a critical number for $f$.




Note: In some texts, endpoints of a closed domain are also considered to be critical numbers. In texts that don't make this definition, the student is instructed to search for extrema at critical numbers and at endpoints.
Theorem 3.7.2 is often called the firstderivative test for a relative extremum, and the content of this theorem is captured by the animations in Figures 3.7.3ad.
Theorem 3.7.2 (FirstDerivative Test)

1.

$f$ is continuous on the closed interval $I$

2.

$c$ is a critical number for $f$

⇒
1.

$f\prime$ changes from positive to negative across $x\=c$ ⇒ $f\left(c\right)$ is a local maximum

2.

$f\prime$ changes from negative to positive across $x\=c$ ⇒ $f\left(c\right)$ is a local minimum

3.

$f\prime$ does not change sign across $x\=c$ ⇒ $f\left(c\right)$ is not a local extremum




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Figures 3.7.3ad animate the passage of a tangent line along the graphs of $1{x}^{2}\,{x}^{2}\,{x}^{3}$, and ${x}^{3}$, respectively. At $x\=0$ the first curve has a maximum; and the second, a minimum. Neither of the last two have an extreme value at the origin, although the slope does go to zero at that point. That is why points where the derivative is zero are only candidates for an extreme value.
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Figure 3.7.3a Maximum





Figure 3.7.3b Minimum





Figure 3.7.3c Neither





Figure 3.7.3d Neither




$a\=$
=



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Some texts will call points where the derivative vanishes stationary points because at such points the function "pauses," or becomes stationary. Some stationary points are extreme points, but some, such as the origin in Figures 3.7.3cd, are not.
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Concavity and Inflection Points


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Another feature of a graph is its concavity, an interval property formalized by Definition 3.7.4.
Definition 3.7.4: Concavity

The graph of $f$ is concave upward on an interval $I$ if it lies above all its tangents on $I$.
Equivalently, if every secant line in $I$ lies above the graph of $f$, then the graph of $f$ is concave upward.
The graph of $f$ is concave downward on an interval $I$ if it lies below all its tangents on $I$.
Equivalently, if every secant line in $I$ lies below the graph of $f$, then the graph of $f$ is concave downward.



The graph in Figure 3.7.3b is concave upward on $\left[1\,1\right]$; the graph in Figure 3.7.3a, downward. In Figure 3.7.3c, the graph is concave downward for $x<0$, but concave upward for $x\>0$. In Figure 3.7.3d, just the opposite happens: the graph is concave upward for $x<0$, but concave downward for $x\>0$. Hence, in Figures 3.7.3cd, the concavity changes at $x\=0$, and such points are called inflection points.
Definition 3.7.5: Inflection Point

The point $\left(a\,f\left(a\right)\right)$ is an inflection point if the concavity of the graph of $f$ changes across $x\=a$.



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Definition 3.7.6, equivalent to Definition 3.7.4, is an alternate way of defining concavity that relates concavity to the behavior of the first derivative.
Definition 3.7.6: Concavity

The graph of a differentiable function $f$ is concave upward on an interval $I$ if $f\prime$ is an increasing function on $I$.
The graph of a differentiable function $f$ is concave downward on an interval $I$ if $f\prime$ is a decreasing function on $I$.



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If $f\prime$ is increasing, then $f\u2033$, its derivative, must be positive. Hence, if $f\u2033\>0$ on an interval, then the graph of $f$ is concave upward on that interval. Correspondingly, if $f\prime$ is decreasing, then $f\u2033$ must be negative. Hence, on an interval where $f\u2033<0$, the graph of $f$ must be concave downward. This leads to the following test for relative extrema.
Theorem 3.7.3 (SecondDerivative Test)

If $f\prime \left(c\right)\=0$ and $f\u2033\left(c\right)<0$ then $f\left(c\right)$ is a relative minimum.
If $f\prime \left(c\right)\=0$ and $f\u2033\left(c\right)\>0$ then $f\left(c\right)$ is a relative maximum.



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Note: There is at least one famous text that uses the term convex for concave upward; and concave, for concave downward.
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Examples


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Example 3.7.1

For the function $f\left(x\right)\=x\mathrm{cos}\left(x\right)$, $x\in \left[0\,2\mathrm{\pi}\right]$ use first principles to obtain the data in Table 3.7.1.

Example 3.7.2

Graph $f\left(x\right)\={x}^{3}\+5{x}^{2}17x9$, $x\in \left[9\,5\right]$; then use the tools of the calculus to analyze the features of this graph.

Example 3.7.3

Graph $f\left(x\right)\={x}^{6}10{x}^{5}15{x}^{4}\+140{x}^{3}\+160{x}^{2}528x800$ for $x\in \left[4\,11\right]$; then use the tools of the calculus to analyze the features of this graph.

Example 3.7.4

Graph $f\left(x\right)\=\frac{65xplus;3{x}^{2}{x}^{3}}{34xplus;{x}^{2}}$ for $x\in \left[5\,10\right]$, indicating any asymptotes this rational function might have. Then, use the tools of the calculus to analyze the features of this graph.

Example 3.7.5

Graph $f\left(x\right)\=\mathrm{sin}\left(4x\+1\/2\right)\mathrm{cos}\left(x\/2\right)$, then use the tools of the calculus to analyze the significant features of this graph.

Example 3.7.6

Determine how the value of $r$ affects the graph of the rational function $f\left(x\right)\=\frac{10}{{x}^{2}\+6xplus;r}$.



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