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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 107c
Chapter 4, Problem 107c

The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>


c. At what point(s) does f have an inflection point?

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Understand that an inflection point occurs where the second derivative changes sign, indicating a change in concavity of the function f.
Since the graph of f' is provided, we need to analyze the behavior of f' to infer information about f''.
Identify where f' has local extrema (maximum or minimum points) on the interval [-3, 2]. These points are where f'' changes sign.
Look for points on the graph of f' where it changes from increasing to decreasing or vice versa. These are potential inflection points for f.
Verify that at these points, the sign of f'' changes by considering the slope of f' before and after these points.

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Key Concepts

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

First Derivative and Critical Points

The first derivative of a function, denoted as f', indicates the rate of change of the function f. Critical points occur where f' is zero or undefined, which can signify local maxima, minima, or points of inflection. Understanding these points is essential for analyzing the behavior of the function and determining where it changes direction.
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Critical Points

Second Derivative and Inflection Points

An inflection point occurs where the concavity of the function changes, which is determined by the second derivative, f''. If f'' changes sign at a point, it indicates a transition in concavity, suggesting that the function f has an inflection point there. Identifying these points is crucial for understanding the overall shape of the graph.
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The Second Derivative Test: Finding Local Extrema

Graphical Interpretation of Derivatives

The graph of the first derivative f' provides valuable insights into the behavior of the original function f. By analyzing where f' is positive or negative, one can infer where f is increasing or decreasing. Additionally, the points where f' crosses the x-axis indicate potential critical points, which are essential for locating inflection points in the context of the original function.
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Graphical Applications of Exponential & Logarithmic Derivatives: Example 8
Related Practice
Textbook Question

Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.

Textbook Question

{Use of Tech} Graph carefully Graph the function f(x) = 60x⁵ - 901x³ + 27x in the window [-4,4] x [-10,000, 10,000]. How many extreme values do you see? Locate all the extreme values by analyzing f'. 

Textbook Question

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>


a. On what interval(s) is f increasing? Decreasing?

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Textbook Question

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

Textbook Question

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Textbook Question

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?