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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 72
Chapter 4, Problem 72

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.


f(x) = x⁴eˣ + x

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1
To determine concavity, we need to find the second derivative of the function. Start by finding the first derivative of f(x) = x⁴eˣ + x. Use the product rule for the term x⁴eˣ and the power rule for x.
The first derivative, f'(x), is found by applying the product rule to x⁴eˣ: f'(x) = d/dx(x⁴)eˣ + x⁴d/dx(eˣ) + d/dx(x). Simplify this expression.
Next, find the second derivative, f''(x), by differentiating f'(x). Again, apply the product rule where necessary and simplify the expression.
Determine the intervals of concavity by setting f''(x) = 0 and solving for x. These solutions will help identify potential inflection points.
Test the intervals around the solutions found in the previous step by choosing test points. Substitute these test points into f''(x) to determine the sign of the second derivative, which indicates whether the function is concave up (f''(x) > 0) or concave down (f''(x) < 0) on those intervals.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Concavity

Concavity refers to the direction in which a function curves. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, it is concave down if the second derivative is negative, meaning the slope is decreasing. Understanding concavity helps in analyzing the behavior of functions and identifying points of inflection.
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Determining Concavity Given a Function

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function and locate inflection points. By calculating the second derivative of a function, we can assess where it changes sign. If the second derivative is positive, the function is concave up; if negative, it is concave down. Inflection points occur where the second derivative equals zero or is undefined, indicating a change in concavity.
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Inflection Points

Inflection points are specific points on a curve where the concavity changes. At these points, the second derivative of the function is either zero or undefined. Identifying inflection points is crucial for understanding the overall shape of the graph and can provide insights into the function's behavior, such as local maxima and minima. They are essential for sketching the graph of a function accurately.
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Critical Points
Related Practice
Textbook Question

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.


f(x) = ³√(x - 4)

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Textbook Question

Another pen problem A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure). <IMAGE>

b. Suppose there is already a fence along the side of the property opposite the side of length y. Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing.

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Textbook Question

The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). <IMAGE>

b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.

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Textbook Question

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.


h(t) = 2 + cos 2t on [0,π]

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Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ log₂ x / log₃ x

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Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ x² ln( cos 1/x)

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