Explanation Test used: A function f with its derivatives f ′ and f ′ ′ exists at all points in an interval ( a , b ) . Then f is concave upward on ( a , b ) if f ′ ′ ( x ) > 0 for all x in ( a , b ) and concave downward on ( a , b ) if f ′ ′ ( x ) < 0 for all x . Definition used: A function f has an inflection point, if the second derivative is 0 or does not exist. Calculation: It is given that the function is f ( x ) = x + cos x . Domain: As the function is defined for any value of x , the domain of the function is ( − ∞ , ∞ ) _ . Critical points: Obtain the first derivative f ′ ( x ) as follows. f ′ ( x ) = d d x ( x + cos x ) = 1 − sin x That is, the first derivative f ′ ( x ) is 1 − sin x . Use f ′ ( c ) = 0 and obtain the critical numbers as follows. 1 − sin c = 0 sin c = 1 sin c = sin ( ( 4 n + 1 ) π 2 ) , n is an integer c = ( 4 n + 1 ) π 2 , n is an integer The value of function f ( x ) = x + cos x at c = ( 4 n + 1 ) π 2 becomes f ( ( 4 n + 1 ) π 2 ) = ( 4 n + 1 ) π 2 + cos ( ( 4 n + 1 ) π 2 ) , n is an integer = ( 4 n + 1 ) π 2 + 0 = ( 4 n + 1 ) π 2 , n is an integer Thus, the critical points are ( ( 4 n + 1 ) π 2 , ( 4 n + 1 ) π 2 ) _ , n is an integer . Symmetry: Obtain the value of f ( − x ) and obtain the result. f ( − x ) = ( − x ) + cos ( − x ) = − x + cos x As neither f ( − x ) = f ( x ) nor f ( − x ) = − f ( x ) , thus there is no symmetry

Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

11th Edition

ISBN: 9780133886832

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13.2, Problem 47E

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To graph: The function with its domain, critical points, symmetry, regions where the function increases or decreases, inflection points, the regions of concave upward or concave downward, and intercepts if exist.

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Chapter 13.2, Problem 46E

Chapter 13.2, Problem 48E

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Chapter 13 Solutions

Calculus with Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Lial, Greenwell & Ritchey, The Applied Calculus & Finite Math Series)

Ch. 13.1 - (a) Convert 210° to radians. (b) Convert 3π/4...Ch. 13.1 - Find the values of the six trigonometric functions...Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Find all values of θ between 0 and 2π that satisfy...Ch. 13.1 - Convert the following degree measures to radians....Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Convert the following degree measures to radians....Ch. 13.1 - Prob. 5E

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