Explanation Given : f ( x ) = ( x + 4 ) ( x − 1 ) 2 x + 3 Concept used: Continuity: The function f is continuous at x = c if f ( c ) is defined and if lim → x → c f ( x ) = f ( c ) . Intermediate value theorem: Intermediate value theorem states that if f is a continuous function whoch domain contains the interval [ a , b ] , then it takes on any given value between f ( a ) and f ( b ) at some point within the interval. Calculation : f ( x ) = ( x + 4 ) ( x − 1 ) 2 x + 3 be a given function. And it has to find the intervals on which the functions is positive or negative. Now roots of function are f ( x ) = ( x + 4 ) ( x − 1 ) 2 x + 3 , 2 x + 3 ≠ 0 ( x + 4 ) ( x − 1 ) = 0 , x ≠ − 3 2 ( x + 4 ) ( x − 1 ) = 0 ( x + 4 ) = 0 or ( x − 1 ) = 0 x = − 4 or x = 1 In each of the intervals between these points the function f is either always positive or always negative. It needs to test the sign of f ( x ) at one point in each interval. Now, f ( x ) = ( x + 4 ) ( x − 1 ) 2 x + 3 f ( − 5 ) = ( − 5 + 4 ) ( − 5 − 1 ) 2 ( − 5 ) + 3 f ( − 5 ) = (

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Question

Chapter 1.4, Problem 70S

To determine

To find:The intervals on which function is positive or negative by using the intermediate value theorem and definition of continuity.

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Chapter 1.4, Problem 69S

Chapter 1.4, Problem 71S

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Chapter 1 Solutions

Calculus

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The intervals on which function is positive or negative by using the intermediate value theorem and definition of continuity.

Solution Summary: The author explains the intermediate value theorem and definition of continuity to find the intervals on which the functions are positive or negative.

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Chapter 1 Solutions