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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.5a
Chapter 4, Problem 4.R.5a

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.

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1
Identify the critical points of the function f by finding where the derivative f' is equal to zero or undefined. These are the x-values where the graph of f' crosses the x-axis or has discontinuities.
Determine the intervals where the function f is increasing by analyzing the sign of f'. If f' is positive on an interval, then f is increasing on that interval.
Determine the intervals where the function f is decreasing by analyzing the sign of f'. If f' is negative on an interval, then f is decreasing on that interval.
Use the second derivative f" to confirm the nature of the critical points. If f" is positive at a critical point, it indicates a local minimum, and if f" is negative, it indicates a local maximum.
Summarize the findings by listing the critical points and specifying the intervals of increase and decrease for the function f, based on the analysis of f' and f".

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Key Concepts

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

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they indicate where the function's slope changes. To find critical points, one typically sets the first derivative, f', equal to zero and solves for the variable.
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Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval where its derivative, f', is positive, and decreasing where f' is negative. By analyzing the sign of the derivative around the critical points, one can determine the intervals of increase and decrease. This information is crucial for understanding the overall behavior of the function.
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Determining Where a Function is Increasing & Decreasing

Second Derivative Test

The second derivative test involves using the second derivative, f'', to determine the concavity of the function at critical points. If f'' is positive at a critical point, the function is concave up, indicating a local minimum; if f'' is negative, the function is concave down, indicating a local maximum. This test helps confirm the nature of critical points found using the first derivative.
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The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = x²/₃(4 -x²) on -2,2

Textbook Question

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

lim_x→∞ (1 - (3/x))ˣ

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Textbook Question

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



c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) cos² x on [0,π]

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Textbook Question

Viewing angles An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 ft above eye level and the upper edge of the screen is 10 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle? <IMAGE>

Textbook Question

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


x¹⸍² and x¹⸍³