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
f and f” are provided below. Use f” to find the open intervals where f is
concave up and concave down.
![The image contains mathematical expressions for a function \( f(x) \) and its second derivative \( f''(x) \).
1. **Function \( f(x) \):**
\[
f(x) = \frac{10x^3}{x^2 - 1}
\]
This expression represents a rational function where the numerator is a polynomial of degree 3 and the denominator is a polynomial of degree 2.
2. **Second Derivative \( f''(x) \):**
\[
f''(x) = \frac{20x(x^2 + 3)}{(x^2 - 1)^3}
\]
This expression represents the second derivative of the function \( f(x) \). The numerator \( 20x(x^2 + 3) \) indicates a product of a linear term and a quadratic term, while the denominator is a cubic function derived from the original denominator raised to the power of 3.
These expressions are useful in calculus for analyzing the behavior and curvature of the function \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5cfc2ea5-360d-45d2-838f-b967e3312bc8%2F0cb46109-fae9-4d77-8f6e-6ce12fd2daf2%2Fb1ns22a_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains mathematical expressions for a function \( f(x) \) and its second derivative \( f''(x) \).
1. **Function \( f(x) \):**
\[
f(x) = \frac{10x^3}{x^2 - 1}
\]
This expression represents a rational function where the numerator is a polynomial of degree 3 and the denominator is a polynomial of degree 2.
2. **Second Derivative \( f''(x) \):**
\[
f''(x) = \frac{20x(x^2 + 3)}{(x^2 - 1)^3}
\]
This expression represents the second derivative of the function \( f(x) \). The numerator \( 20x(x^2 + 3) \) indicates a product of a linear term and a quadratic term, while the denominator is a cubic function derived from the original denominator raised to the power of 3.
These expressions are useful in calculus for analyzing the behavior and curvature of the function \( f(x) \).
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