Explanation Given: The polynomial is f ( x ) = ( x − 6 ) ( x + 6 ) 2 . Calculation: Domain: In order to calculate the domain, take any value of x in the given polynomial, which is a real number. Hence, clearly the domain of f is ( − ∞ , ∞ ) . Symmetry: Obtain the symmetry of the given polynomial. Substitute the value f ( x ) = f ( − x ) . That is, f ( − x ) = ( − x − 6 ) ( − x + 6 ) 2 = − ( x + 6 ) ( x − 6 ) 2 This is neither odd symmetry nor even symmetry. Therefore, the polynomial has no symmetry. Intercept: Let f ( x ) = 0 . That is ( x − 6 ) ( x + 6 ) 2 = 0 That is x intercept is 6 and -6 and the y intercept is f ( 0 ) = − 216 . Derivatives and possible inflections: From the given function is f ( x ) = ( x − 6 ) ( x + 6 ) 2 = ( x − 6 ) ( x 2 + 36 + 12 x ) = x 3 + 36 x + 12 x 2 − 6 x 2 − 216 − 72 x = x 3 + 6 x 2 − 36 x − 216 . Obtain the derivative of the given function. That is, f ′ ( x ) = 3 x 2 + 12 x − 36 . Which is 0 when x = 2 or -6. Thus, the possible inflection point occurs at 0 when x = 2 or -6. Increasing / decreasing and concavity: Assume the value of x = − 7 . That is, f ′ ( − 7 ) = 3 ( − 7 ) 2 + 12 ( − 7 ) − 36 = 3 ⋅ 49 − 84 − 36 = 147 − 84 − 36 = 27 Which is greater than zero. Take the value x = − 2 . That is, f ′ ( − 2 ) = 3 ( − 2 ) 2 + 12 ( − 2 ) − 36 = 12 − 24 − 36 = − 48 Which is less than zero

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

2nd Edition

ISBN: 9780321954237

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 4.3, Problem 13E

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To identify: The local extrema, inflection points, and x and y intercepts when the given polynomial exists; sketch a graph for the given polynomial.

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Chapter 4.3, Problem 12E

Chapter 4.3, Problem 14E

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Chapter 4 Solutions

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

Ch. 4.1 - Prob. 1QC Ch. 4.1 - Prob. 2QC Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Sketch the graph of a function that has an...Ch. 4.1 - What is a critical point of a function?Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...

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The local extrema, inflection points, and x and y intercepts when the given polynomial exists; sketch a graph for the given polynomial.

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To identify: The local extrema, inflection points, and x and y intercepts when the given polynomial exists; sketch a graph for the given polynomial.

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Chapter 4 Solutions