(a) Explanation Given Information: The function is f ( x ) = sin x − cos x and the interval is 0 ≤ x ≤ 2 π . Calculation: Point of local maximum . The point at which the derivative of function changes positive value to negative value is called local maximum point. Point of local minimum . The point at which the derivative of function changes negative value to positive value is called local minimum point. Provide the function as follows. f ( x ) = sin x − cos x (1) Calculate the first derivative of f ( x ) . f ′ ( x ) = cos x + sin x (2) Calculate the critical point of function f . At the critical point, the first derivative of f is zero. cos x + sin x = 0 cos x = − sin x tan x = − 1 x = 3 π 4 and x = 7 π 4 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. The sign pattern of reveals the behavior of f between and at the critical points, as summarized in the Table 2. Interval [ 0 , 3 π 4 ] [ 3 π 4 , 7 π 4 ] [ 7 π 4 , 2 π ] Sign of f ′ ( x ) + ‑ + Check the function values both critical point and end points (b) To determine Graph the function and the derivative of the function in the given domain and comment the behaviour of graph.

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

4th Edition

ISBN: 9780136880912

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

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Chapter 4.3, Problem 60E

(a)

To determine

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

(b)

To determine

Graph the function and the derivative of the function in the given domain and comment the behaviour of graph.

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Chapter 4.3, Problem 59E

Chapter 4.3, Problem 61E

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Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

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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