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) = 4x¹⸍² - x⁵⸍² on [0, 4]
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.6
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]
Verified step by step guidance1
First, find the derivative of the function ƒ(x) = x³ - 6x². The derivative, ƒ'(x), will help us identify the critical points. Use the power rule to differentiate: ƒ'(x) = 3x² - 12x.
Set the derivative ƒ'(x) = 3x² - 12x equal to zero to find the critical points. Solve the equation 3x² - 12x = 0 for x. Factor the equation: 3x(x - 4) = 0, which gives the solutions x = 0 and x = 4.
Evaluate the function ƒ(x) at the critical points and at the endpoints of the interval [-1, 5]. Calculate ƒ(-1), ƒ(0), ƒ(4), and ƒ(5) to determine the function values at these points.
Compare the values of ƒ(x) at the critical points and endpoints to identify the absolute maximum and minimum values on the interval [-1, 5]. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum.
Conclude by stating the absolute maximum and minimum values, if they exist, based on the evaluations from the previous step. Ensure to check that these values fall within the given interval.
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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 either 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 these conditions.
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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 that 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 indicate the absolute maximum and minimum, respectively.
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Closed Interval
A closed interval, denoted as [a, b], includes all numbers between a and b, as well as the endpoints a and b themselves. This is significant in calculus because the Extreme Value Theorem states that a continuous function on a closed interval must attain both an absolute maximum and minimum. Thus, when analyzing functions over closed intervals, one must consider the behavior of the function at both the endpoints and any critical points within the interval.
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Related Practice
Textbook Question
45–46. Linear approximation
a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?
ƒ(x) = x²⸍³ ; a =27; ƒ(29)
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).
g(x) = x sin⁻¹ x on [-1, 1]
1
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Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
Textbook Question
104–107. Functions from derivatives Find the function f with the following properties.
ƒ'(t) = sin t + 2t; ƒ(0) = 5
Textbook Question
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
eˣ and 3ˣ