Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ ― 2𝓍 + 4
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 4.3.17
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
Verified step by step guidance1
Observe the graph of the function y = f(x) and identify the critical points where the function changes direction. These points are typically where the derivative is zero or undefined.
Determine the intervals where the function is increasing. A function is increasing on an interval if its derivative is positive over that interval. From the graph, identify the segments where the slope of the function is positive.
Determine the intervals where the function is decreasing. A function is decreasing on an interval if its derivative is negative over that interval. From the graph, identify the segments where the slope of the function is negative.
Identify the local extrema by examining the critical points. A local maximum occurs where the function changes from increasing to decreasing, and a local minimum occurs where the function changes from decreasing to increasing.
Identify the absolute extrema by comparing the values of the function at the critical points and endpoints of the interval. The highest value is the absolute maximum, and the lowest value is the absolute minimum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Increasing and Decreasing Intervals
A function is increasing on an interval if, for any two points within the interval, a larger x-value results in a larger y-value. Conversely, it is decreasing if a larger x-value results in a smaller y-value. To determine these intervals, one can analyze the graph's slope or use the first derivative test, where a positive derivative indicates increasing and a negative derivative indicates decreasing.
Recommended video:
07:32
Determining Where a Function is Increasing & Decreasing
Local and Absolute Extrema
Local extrema are points where a function reaches a local maximum or minimum within a neighborhood, while absolute extrema are the highest or lowest points over the entire domain. Local extrema occur where the derivative changes sign, and absolute extrema can be found by evaluating the function at critical points and endpoints. The graph helps visualize these points as peaks (maxima) or troughs (minima).
Recommended video:
05:58Finding Extrema Graphically
Critical Points
Critical points of a function occur where its derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. These points are crucial for identifying intervals of increase or decrease and locating extrema. By analyzing the graph or calculating the derivative, one can pinpoint these critical points and assess their nature using the first or second derivative tests.
Recommended video:
04:50Critical Points
Related Practice
Textbook Question
Applications
Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?
Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = (x² − 3) / (x − 2), x ≠ 2
Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = √(x(1 − x)), [0, 1]
Textbook Question
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
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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.
∫(t√t + √t) / t² dt