(a) Explanation Calculation: Write the given derivative as follows. K ( t ) = 15 t 3 − t 5 (1) Find the first derivative of K ( t ) . K ′ ( t ) = 15 ( 3 t 2 ) − 5 t 4 = 45 t 2 − 5 t 4 (2) Find the critical point of function H . At the critical point, the first derivative of H is zero. 45 t 2 − 5 t 4 = 0 5 t 2 ( 9 − t 2 ) = 0 (3) The root of Equation (3) is the critical point of function K . Find the first root of Equation (3). 5 t 2 = 0 t = 0 Find the second root of Equation (3). ( 9 − t 2 ) = 0 t 2 = 9 t = ± 3 The critical points of K is t = 0 , t = − 3 , and t = 3 . The behaviour of K according to sign pattern of K ′ as shown in Table 1. Sign of K ′ Behaviour of K + (Positive) Increasing ‑ (Negative) Decreasing Testing all intervals to the left and right of the critical point. Write the intervals of the function K using critical points. ( − ∞ , − 3 ) ( − 3 , 0 ) ( 0 , 3 ) ( 3 , ∞ ) Consider an interval ( − ∞ , − 3 ) . Substitute any value for t in the interval ( − ∞ , − 3 ) in Equation (2). Substitute ‑4 for t in Equation (2). K ′ ( − 4 ) = 45 ( − 4 ) 2 − 5 ( − 4 ) 4 = 720 − 1280 = − 560 Consider an interval ( − 3 , 0 ) (b) To determine Find the function’s local extreme values and the point of occurrence.

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ISBN: 9780135910993

Author: Hass

Publisher: PEARSON

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Chapter 4.3, Problem 30E

(a)

To determine

Find the increasing and decreasing open interval of function K.

(b)

To determine

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

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Chapter 4.3, Problem 29E

Chapter 4.3, Problem 31E

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MYLAB MATH W/PEARSON ETEXT 18 WEEK CODE

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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Find the increasing and decreasing open interval of function K .