Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ - 6x² on [-1, 5]
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.12
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = sin 2x + 3 on [-π , π]
Verified step by step guidance1
To find the critical points of the function ƒ(x) = sin(2x) + 3 on the interval [-π, π], first compute the derivative of the function. The derivative of sin(2x) is 2cos(2x), so the derivative of ƒ(x) is ƒ'(x) = 2cos(2x).
Set the derivative ƒ'(x) = 2cos(2x) equal to zero to find the critical points. Solve the equation 2cos(2x) = 0, which simplifies to cos(2x) = 0.
The solutions to cos(2x) = 0 are 2x = π/2 + kπ, where k is an integer. Solve for x to find x = π/4 + kπ/2.
Determine which solutions for x fall within the interval [-π, π]. Substitute k = -2, -1, 0, 1, 2 into x = π/4 + kπ/2 to find the valid critical points within the interval.
Evaluate the function ƒ(x) = sin(2x) + 3 at the critical points and at the endpoints x = -π and x = π to determine the absolute maximum and minimum values. Compare these values to identify the absolute extrema.
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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 represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy the condition.
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Critical Points
Absolute Maximum and Minimum
The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval. To determine these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest and smallest of these values will be the absolute maximum and minimum, respectively.
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Interval Notation
Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [-π, π] indicates that the function is being analyzed from -π to π, including both endpoints. Understanding interval notation is crucial for correctly identifying where to evaluate the function and its critical points.
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Related Practice
Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))
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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
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24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ
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Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = ln( x² + 3) / (x -1)
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Textbook Question
Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>
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