Explanation Given: The function is f ( x ) = 2 cos ( 3 x ) + 3 cos ( 2 x ) . Concept used: Critical points: For a function f ( x ) the critical points exists at such points such that the value of f ′ ( x ) is 0 or the f ′ ( x ) does not exists. Calculation: The given function is expressed as follows: f ( x ) = 2 cos ( 3 x ) + 3 cos ( 2 x ) …..(1) Differentiate equation (1) as shown below: f ′ ( x ) = d ( 2 cos ( 3 x ) ) d x + d ( 3 cos ( 2 x ) ) d x = − 6 sin ( 3 x ) − 6 sin ( 2 x ) To obtain the critical points for the given function solve the equation f ′ ( x ) = 0 as shown below: f ′ ( x ) = 0 − 6 sin ( 3 x ) − 6 sin ( 2 x ) = 0 sin ( 3 x ) + sin ( 2 x ) = 0 2 sin ( 3 x + 2 x 2 ) cos ( 3 x − 2 x 2 ) = 0 Further solve the above equation. 2 sin ( 3 x + 2 x 2 ) cos ( 3 x − 2 x 2 ) = 0 sin ( 5 x 2 ) cos ( x 2 ) = 0 Use the zero product property to solve the above equation. First solve the equation sin ( 5 x 2 ) = 0 as shown below: sin ( 5 x 2 ) = 0 sin ( 5 x 2 ) = sin ( n π ) 5 x 2 = n π x = 2 n π 5 The value of x is expressed as shown below: x = 2 n π 5 ….(2) In the above equation n = 0 , 1 , 2 , 3... . Now, solve the equation cos ( x 2 ) = 0 as shown below: cos ( x 2 ) = 0 cos ( x 2 ) = cos ( ( 2 n + 1 ) π 2 ) x 2 = ( 2 n + 1 ) π 2 x = ( 2 n + 1 ) π The value of x is expressed as shown below: x = ( 2 n + 1 ) π …

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 4.2, Problem 64E

To determine

To find:

The critical points of the function f(x)=2cos(3x)+3cos(2x) in the interval [0,2π] and check the answer with the graph of the function f(x).

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Chapter 4.2, Problem 63E

Chapter 4.2, Problem 65E

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Chapter 4 Solutions

Calculus: Early Transcendentals

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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The critical points of the function f ( x ) = 2 cos ( 3 x ) + 3 cos ( 2 x ) in the interval [ 0 , 2 π ] and check the answer with the graph of the function f ( x ) .

Solution Summary: The author analyzes the critical points of a function f(x).

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To find:

The critical points of the function f(x)=2cos(3x)+3cos(2x) in the interval [0,2π] and check the answer with the graph of the function f(x).

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Chapter 4 Solutions

The critical points of the function f ( x ) = 2 cos ( 3 x ) + 3 cos ( 2 x ) in the interval [ 0 , 2 π ] and check the answer with the graph of the function f ( x ) .

Solution Summary: The author analyzes the critical points of a function f(x).

Concept explainers

Videos

To find: The critical points of the function f(x)=2cos(3x)+3cos(2x) in the interval [0,2π] and check the answer with the graph of the function f(x).

Want to see the full answer?

Chapter 4 Solutions

To find:

The critical points of the function f(x)=2cos(3x)+3cos(2x) in the interval [0,2π] and check the answer with the graph of the function f(x).