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Let f(x) = · – 3(x + 4)"(x – 4)1º(x + 5)"| |

Let f(x) = · – 3(x + 4)"(x – 4)1º(x + 5)"| |

Calculus: Early Transcendentals
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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2 d 

 

Explain and Demonstrate how to solve f(x)>0

The function \( f(x) = -3(x+4)^{13}(x-4)^{10}(x+5)^{11} \) is defined. This expression represents a polynomial function, which is the product of three binomial terms, each raised to an exponent. The polynomial is multiplied by a constant factor of \(-3\). The terms within the function include: 1. \( (x+4)^{13} \): This term indicates that the binomial \( (x+4) \) is raised to the 13th power, suggesting it will contribute 13 roots located at \( x = -4 \). 2. \( (x-4)^{10} \): Here, the binomial \( (x-4) \) is raised to the 10th power, indicating 10 roots at \( x = 4 \). 3. \( (x+5)^{11} \): This indicates 11 roots at \( x = -5 \) due to the binomial \( (x+5) \) being raised to the 11th power. The negative sign in front of the constant factor \(-3\) suggests that the graph of this polynomial will be reflected across the x-axis. This function is an example of a higher-degree polynomial with multiplicities of roots; such characteristics will affect its graph's shape and behavior around the roots.
Transcribed Image Text:The function \( f(x) = -3(x+4)^{13}(x-4)^{10}(x+5)^{11} \) is defined. This expression represents a polynomial function, which is the product of three binomial terms, each raised to an exponent. The polynomial is multiplied by a constant factor of \(-3\). The terms within the function include: 1. \( (x+4)^{13} \): This term indicates that the binomial \( (x+4) \) is raised to the 13th power, suggesting it will contribute 13 roots located at \( x = -4 \). 2. \( (x-4)^{10} \): Here, the binomial \( (x-4) \) is raised to the 10th power, indicating 10 roots at \( x = 4 \). 3. \( (x+5)^{11} \): This indicates 11 roots at \( x = -5 \) due to the binomial \( (x+5) \) being raised to the 11th power. The negative sign in front of the constant factor \(-3\) suggests that the graph of this polynomial will be reflected across the x-axis. This function is an example of a higher-degree polynomial with multiplicities of roots; such characteristics will affect its graph's shape and behavior around the roots.
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