Explanation Given: The given function is, f ( x ) = x + cos x Approach: The critical points ( x , f ( x ) ) can be find by substituting 0 for f ′ ( x ) and finding value of x and substituting those values in the function. The inflection points of a function are the points ( x , f ( x ) ) for which f ″ ( x ) = 0 . The x intercept of the function are the points ( x , f ( x ) ) for which f ( x ) = 0 . The y intercept of the function are the points ( 0 , f ( 0 ) ) for which x = 0 . Calculation: Consider the given function. f ( x ) = x + cos x ... ( 1 ) The above function is a polynomial function. So it has no asymptotes and its domain is ℝ . Differentiate the above function with respect to x . f ( x ) = 1 − sin x Substitute 0 for f ( x ) in above equation to find critical points. 1 − sin x = 0 sin x = 1 x = ( 2 n + 1 2 ) π ; n ∈ ℤ Substitute ( 2 n + 1 2 ) π for x in equation ( 1 ) . f ( ( 2 n + 1 2 ) π ) = ( 2 n + 1 2 ) π + cos ( ( 2 n + 1 2 ) π ) = ( 2 n + 1 2 ) π So, the critical point of the function is ( ( 2 n + 1 2 ) π , ( 2 n + 1 2 ) π ) . Differentiate equation ( 1 ) two times with respect to x . f ″ ( x ) = − cos x Substitute 0 for f ″ ( x ) in above equation to find inflection points. cos x = 0 x = ( n + 1 2 ) π ; n ∈ ℤ Substitute ( n + 1 2 ) π for x in equation ( 1 )

2nd Edition

ISBN: 9780321964038

Author: GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher: Pearson Addison Wesley,

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Chapter 5.4, Problem 31E

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The function f(x)=x+cosx considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

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Chapter 5.4, Problem 30E

Chapter 5.4, Problem 32E

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To graph: The function f ( x ) = x + cos x considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

Solution Summary: The author explains that the given function is a polynomial function. It has no asymptotes and its domain is R.

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To graph:

The function f(x)=x+cosx considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

Want to see the full answer?

Chapter 5 Solutions

To graph: The function f ( x ) = x + cos x considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

Solution Summary: The author explains that the given function is a polynomial function. It has no asymptotes and its domain is R.

Concept explainers

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To graph: The function f(x)=x+cosx considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

Want to see the full answer?

Chapter 5 Solutions

To graph:

The function f(x)=x+cosx considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.