Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Find the absolute maximum and absolute minimum values of the function on the indicated interval. (second term is three x to the two thirds power)
![### Function Definition
The given function is defined as:
\[ f(x) = x^2 - 3x^{2/3} \]
### Domain
The function is evaluated over the interval \([0, 2]\).
### Explanation
- **\(x^2\)**: This term represents a standard quadratic term, which is a parabolic curve opening upwards.
- **\(3x^{2/3}\)**: This term is a fractional power of \(x\), and represents a curve that grows slower than a quadratic function. The \(2/3\) exponent modifies how steeply or shallowly the function grows as \(x\) increases.
- **Subtraction**: The function \(f(x)\) is the result of subtracting the \(3x^{2/3}\) component from the \(x^2\) component, influencing the shape and the position of the curve.
### Interval Analysis
The function is analyzed within the range of \(x\) values from 0 to 2, which affects how the components interact and create the resulting curve within this bounded region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5cfc2ea5-360d-45d2-838f-b967e3312bc8%2F9a7cf89c-f779-4f14-9c50-b19ee38062a0%2Fg37pqrd_processed.png&w=3840&q=75)
Transcribed Image Text:### Function Definition
The given function is defined as:
\[ f(x) = x^2 - 3x^{2/3} \]
### Domain
The function is evaluated over the interval \([0, 2]\).
### Explanation
- **\(x^2\)**: This term represents a standard quadratic term, which is a parabolic curve opening upwards.
- **\(3x^{2/3}\)**: This term is a fractional power of \(x\), and represents a curve that grows slower than a quadratic function. The \(2/3\) exponent modifies how steeply or shallowly the function grows as \(x\) increases.
- **Subtraction**: The function \(f(x)\) is the result of subtracting the \(3x^{2/3}\) component from the \(x^2\) component, influencing the shape and the position of the curve.
### Interval Analysis
The function is analyzed within the range of \(x\) values from 0 to 2, which affects how the components interact and create the resulting curve within this bounded region.
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