Derivative Graph Of A Function And Its Derivative, Derivatives — Learn Desmos

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.

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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,

*

has a critical point at since

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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’ Figure 1. Both functions are increasing over the interval At each point the derivative

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0.” title=”Rendered by QuickLaTeX.com” height=”18″ width=”76″ style=”vertical-align: -4px;” /> Both functions are decreasing over the interval At each point the derivative

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*

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 changes sign as increases through If is differentiable at the only way that

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can change sign as increases through is if

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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 or is undefined. Consequently, to locate local extrema for a function

*

we look for points in the domain of such that or 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 switches as increases through that point.

0. Then, f decreases from x = a to x = b (so f’ 0. The function has an inversion point at c, and it is marked f’(c) = 0. The function increases some more to d (so f’ > 0), which is the global maximum. It is marked that f’(d) = 0. Then the function decreases and it is marked that f’ > 0.” width=”867″ height=”429″> Figure 2. The function has four critical points:

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The function has local maxima at

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and

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and a local minimum at

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The function does not have a local extremum at The sign of changes at all local extrema.

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

If a continuous function has a local extremum, it must occur at a critical point The function has a local extremum at the critical point if and only if the derivative switches sign as increases through Therefore, to test whether a function has a local extremum at a critical point we must determine the sign of to the left and right of

This result is known as the first derivative test.

First Derivative Test

Suppose that is a continuous function over an interval containing a critical point If is differentiable over except possibly at point then satisfies one of the following descriptions:

If changes sign from positive when

*

Problem-Solving Strategy: Using the First Derivative Test

Consider a function that is continuous over an interval

Find all critical points of and divide the interval into smaller intervals using the critical points as endpoints.Analyze the sign of in each of the subintervals. If is continuous over a given subinterval (which is typically the case), then the sign of in that subinterval does not change and, therefore, can be determined by choosing an arbitrary test point in that subinterval and by evaluating the sign of at that test point. Use the sign analysis to determine whether is increasing or decreasing over that interval.

Use the first derivative test to find the location of all local extrema for

*

Use a graphing utility to confirm your results.

Solution

Step 1. The derivative is

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To find the critical points, we need to find where

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Factoring the polynomial, we conclude that the critical points must satisfy

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Therefore, the critical points are

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Now divide the interval into the smaller intervals

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Step 2. Since is a continuous function, to determine the sign of over each subinterval, it suffices to choose a point over each of the intervals

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*

and determine the sign of at each of these points. For example, let’s choose

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as test points.

IntervalTest PointSign of

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at Test PointConclusion

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is increasing.

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*
*
is decreasing.
*
*
*
is increasing.

Step 3. Since switches sign from positive to negative as increases through

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has a local maximum at Since switches sign from negative to positive as increases through

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has a local minimum at

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These analytical results agree with the following graph.

Figure 3. The function has a maximum at

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and a minimum at

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Use the first derivative test to locate all local extrema for

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Solution

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

Hint

Find all critical points of and determine the signs of over particular intervals determined by the critical points.

Use the first derivative test to find the location of all local extrema for

*

Use a graphing utility to confirm your results.

*

The derivative when

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Therefore, at

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The derivative is undefined at Therefore, we have three critical points:

*

and Consequently, divide the interval into the smaller intervals

*

and

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Step 2: Since is continuous over each subinterval, it suffices to choose a test point in each of the intervals from step 1 and determine the sign of at each of these points. The points

*

are test points for these intervals.

IntervalTest PointSign of

*

at Test PointConclusion

is decreasing.
*
is increasing.
*
is increasing.
*
*
is decreasing.

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Step 3: Since is decreasing over the interval and increasing over the interval

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has a local minimum at Since is increasing over the interval and the interval

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does not have a local extremum at Since is increasing over the interval and decreasing over the interval

*

has a local maximum at

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The analytical results agree with the following graph.

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