Textbook Question
Finding Extrema from Graphs
In Exercises 7–10, find the absolute extreme values and where they occur.
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.62
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
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Examine the graph of f'. The graph shows horizontal lines, indicating constant values of f' over intervals.
Identify the intervals where f' is positive, negative, or zero. Positive values of f' suggest f is increasing, negative values suggest f is decreasing, and zero values suggest potential extrema.
Look for points where f' changes sign. These points are candidates for local extrema. A change from positive to negative indicates a local maximum, while a change from negative to positive indicates a local minimum.
In the graph, f' is positive for x < 0, zero at x = 0, and negative for x > 0. This suggests a local maximum at x = 0.
Verify the continuity of f at the identified points. Since f is continuous, the behavior of f' at these points confirms the presence of local extrema.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Critical Points
The derivative of a function, denoted as f', represents the rate of change of the function f. Critical points occur where the derivative is zero or undefined, indicating potential locations for local maxima or minima. Analyzing these points is essential for identifying where the function changes from increasing to decreasing or vice versa.
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Critical Points
First Derivative Test
The First Derivative Test is a method used to determine the nature of critical points. If f' changes from positive to negative at a critical point, it indicates a local maximum; if it changes from negative to positive, it indicates a local minimum. This test helps in classifying the extrema based on the behavior of the derivative around the critical points.
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07:09The First Derivative Test: Finding Local Extrema
Continuity of the Function
Continuity of a function f is crucial for applying the concepts of calculus effectively. A continuous function does not have breaks, jumps, or holes, ensuring that the behavior of f' accurately reflects the behavior of f. This property guarantees that local extrema identified through the derivative correspond to actual extrema in the function itself.
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05:34Intro to Continuity
Related Practice
Textbook Question
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
Textbook Question
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
2. y=x^4/4-2x^2+4
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
f(t) = 2 − |t|, −1 ≤ t ≤ 3
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
77. y' = cot(θ/2), for 0 < θ < 2π
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(1 + cos 4t)/2 dt