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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 4.4.7a
Chapter 4, Problem 4.4.7a

Sketch a graph of a function f with the following properties.


f' < 0 and f" < 0, for x < 3

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1
Understand the given conditions: f'(x) < 0 implies the function f is decreasing, and f''(x) < 0 implies the function is concave down for x < 3.
Start by sketching the graph for x < 3. Since f'(x) < 0, the graph should slope downward as x approaches 3. This indicates a decreasing function.
Next, incorporate the concavity condition f''(x) < 0. Concave down means the graph curves downward, resembling a 'frown' shape. Ensure the graph bends downward as it decreases.
Choose a point at x = 3 to mark the boundary where the given conditions apply. The behavior of the graph beyond x = 3 is not specified, so focus only on the region x < 3.
Label the axes and ensure the graph visually represents both decreasing behavior and concave down curvature for x < 3. This completes the sketch based on the given properties.

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Key Concepts

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

First Derivative Test

The first derivative of a function, denoted as f', indicates the slope of the tangent line to the graph at any point. If f' < 0, the function is decreasing in that interval. Understanding this concept is crucial for sketching the graph, as it informs us that the function is moving downward for x < 3.
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The First Derivative Test: Finding Local Extrema

Second Derivative Test

The second derivative, denoted as f'', provides information about the concavity of the function. If f'' < 0, the function is concave down, meaning that the slope of the tangent line is decreasing. This concept is essential for determining the curvature of the graph, indicating that the function is bending downwards for x < 3.
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The Second Derivative Test: Finding Local Extrema

Graph Behavior

Understanding how the first and second derivatives affect the graph's behavior is key to sketching it accurately. With f' < 0 and f'' < 0 for x < 3, the graph will not only be decreasing but also concave down, suggesting a continuous downward slope that becomes steeper as x approaches 3. This overall behavior shapes the visual representation of the function.
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Graphing The Derivative
Related Practice
Textbook Question

Folded boxes


a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

Textbook Question

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


∫(1 + 3 cosΘ) dΘ

Textbook Question

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.


a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


2ˣ and 4ˣ⸍²

Textbook Question

{Use of Tech} Investment problem A one-time investment of \(2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is a(r) = 2500(1 + r/12)⁶⁰. 


a. Use Newton’s method to find the value of r if the goal is to have \)3200 in the account after 5 years.

Textbook Question

Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.


a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.