(a) To determine Find the increasing and decreasing open interval of function f . Explanation Calculation: Write the given derivative as follows. f ( x ) = x 3 3 x 2 + 1 (1) Find the first derivative of f ( x ) . f ′ ( x ) = 3 x 2 ( 3 x 2 + 1 ) − x 3 ( 6 x ) ( 3 x 2 + 1 ) 2 = 9 x 4 + 3 x 2 − 6 x 4 ( 3 x 2 + 1 ) 2 = 3 x 4 + 3 x 2 ( 3 x 2 + 1 ) 2 = 3 x 2 ( x 2 + 1 ) ( 3 x 2 + 1 ) 2 (2) Find the critical point of function f . At the critical point, the first derivative of f is zero. 3 x 2 ( x 2 + 1 ) ( 3 x 2 + 1 ) 2 = 0 (3) The root of Equation (3) is the critical point of function f . Find the root of Equation (3). 3 x 2 = 0 x = 0 For ( x 2 + 1 ) and ( 3 x 2 + 1 ) 2 , x is an undefined one. The critical points of f is x = 0 . The behaviour of f according to sign pattern of f ′ as shown in Table 1. Sign of f ′ Behaviour of f + (Positive) Increasing ‑ (Negative) Decreasing Testing all intervals to the left and right of the critical point. Write the intervals of the function f using critical points. ( − ∞ , 0 ) ( 0 , ∞ ) Consider an interval ( − ∞ , 0 ) (b) To determine Find the function’s local extreme values and the point of occurrence.

University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)

3rd Edition

ISBN: 9780321999610

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 4.3, Problem 36E

(a)

To determine

Find the increasing and decreasing open interval of function f.

(b)

To determine

Find the function’s local extreme values and the point of occurrence.

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Chapter 4.3, Problem 35E

Chapter 4.3, Problem 37E

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Chapter 4 Solutions

University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)

Ch. 4.1 - In Exercises 16, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 3E Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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