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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 4.R.2c
Chapter 4, Problem 4.R.2c

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
c. Give the approximate coordinates of the inflection point(s) of f.

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Understand that an inflection point is where the function changes concavity, which means the second derivative changes sign.
Examine the graph visually to identify where the curve changes from concave up to concave down or vice versa.
Look for points on the graph where the slope of the tangent line changes from increasing to decreasing or vice versa, indicating a change in concavity.
Estimate the x-coordinate of the inflection point by observing the graph and noting where this change occurs.
Approximate the y-coordinate by finding the corresponding value of the function at the estimated x-coordinate.

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Key Concepts

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

Inflection Points

Inflection points are points on the graph of a function where the curvature changes direction. This means that the second derivative of the function changes sign at these points. Identifying inflection points is crucial for understanding the behavior of the function, particularly in determining where it transitions from concave up to concave down or vice versa.
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Critical Points

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function at a given point. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down. An inflection point occurs where the second derivative equals zero or is undefined, indicating a potential change in concavity.
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The Second Derivative Test: Finding Local Extrema

Graphical Analysis

Graphical analysis involves examining the visual representation of a function to identify key features such as extrema, inflection points, and intervals of increase or decrease. By analyzing the graph, one can estimate the coordinates of inflection points and understand the overall behavior of the function across the specified interval.
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Determining Differentiability Graphically
Related Practice
Textbook Question

{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

Textbook Question

{Use of Tech} Newton’s method Use Newton’s method to approximate the roots of ƒ(x) = e⁻²ˣ + 2eˣ - 6 to six digits.

Textbook Question

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.



b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).

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Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 


lim_x→0 csc x sin⁻¹ x

Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = x³ ln x on (0, ∞)

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Textbook Question

90–103. Indefinite integrals Determine the following indefinite integrals.


∫ dx / (1 - sin² x)