Textbook Question
104–107. Functions from derivatives Find the function f with the following properties.
h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3)
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.14
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]
Verified step by step guidance1
To find the critical points of the function \( f(x) = 4x^{3/2} - x^{5/2} \), first compute the derivative \( f'(x) \). Use the power rule: \( \frac{d}{dx}[x^n] = nx^{n-1} \).
Calculate \( f'(x) = \frac{d}{dx}[4x^{3/2}] - \frac{d}{dx}[x^{5/2}] = 6x^{1/2} - \frac{5}{2}x^{3/2} \).
Set \( f'(x) = 0 \) to find the critical points: \( 6x^{1/2} - \frac{5}{2}x^{3/2} = 0 \). Factor out \( x^{1/2} \): \( x^{1/2}(6 - \frac{5}{2}x) = 0 \). Solve for \( x \).
The solutions to \( x^{1/2}(6 - \frac{5}{2}x) = 0 \) are \( x = 0 \) and \( 6 - \frac{5}{2}x = 0 \). Solve \( 6 - \frac{5}{2}x = 0 \) to find \( x = \frac{12}{5} \).
Evaluate \( f(x) \) at the critical points \( x = 0 \), \( x = \frac{12}{5} \), and the endpoints \( x = 0 \) and \( x = 4 \) to determine the absolute maximum and minimum values on the interval \([0, 4]\).
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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 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 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 represent the absolute maximum and minimum, respectively.
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Evaluating Functions on Intervals
When analyzing functions on a specific interval, it is crucial to evaluate the function at both the endpoints and any critical points found within the interval. This process ensures that all potential maximum and minimum values are considered. The function's behavior can vary significantly across different intervals, making this evaluation vital for accurate conclusions.
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Related Practice
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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Textbook Question
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.
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
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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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