Explanation Given information :. f ( x ) = sin ( x 4 ) Formula Used: Power rule of derivative: If f ( x ) = k x n then f ' ( x ) = k ⋅ n x n − 1 Chain rule: ( f ( g ( x ) ) ) ' = f ' ( g ( x ) ) ⋅ g ' ( x ) Concept Involved: Find the first derivative and second derivative of the given function. Graph f , f ' , & f ' ' In the function f graph, the interval where f < 0 {below the x -axis} is the interval where the function negative and the interval where f > 0 {above the x -axis} is the interval where the function positive. In the first derivative of function f ' graph, the interval where f ' < 0 {below the x -axis} is the interval where the function decreasing and the interval where f ' > 0 {above the x -axis} is the interval where the function increasing.Critical number is the value of x where the f ' ( x ) = 0 . In the second derivative of function f ' ' graph, the interval where f ' ' < 0 {below the x -axis} is the interval where the function concave down and the interval where f ' ' > 0 {above the x -axis} is the interval where the function concave up. Calculation: Description Steps The function will be positive in the interval where f ( x ) > 0 sin ( x 4 ) > 0 ( 8 π n , 8 π n + 4 π ) , n ∈ ℤ The function will be negative in the interval where f ( x ) < 0 sin ( x 4 ) < 0 ( 8 π n − 4 π , 8 π n ) , n ∈ ℤ Find the derivative of the function f ( x ) = sin ( x 4 ) f ' ( x ) = cos ( x 4 ) ⋅ d d x ( x 4 ) = 1 4 cos ( x 4 ) The critical number is the value of x that makes the first derivative f ' ( x ) = 1 4 cos ( x 4 ) equal to zero or undefined. 1 4 cos ( x 4 ) = 0 Solving the equation 1 4 cos ( x 4 ) = 0 by Multiplying 4 on both sides, then Taking arccosine on both sides Then Multiplying 4 on both sides 4 ⋅ 1 4 cos ( x 4 ) = 4 ⋅ 0 cos ( x 4 ) = 0 cos − 1 ( cos ( x 4 ) ) = cos − 1 ( 0 ) 4 ⋅ x 4 = 4 ⋅ cos − 1 ( 0 ) x = 4 cos − 1 ( 0 ) x = 4 ( π n − π 2 ) , n ∈ ℤ n = 4 π n − 2 π , n ∈ ℤ List the values of x {the critical numbers} that makes the first derivative of given function as zero or undefined n = 4 π n − 2 π , n ∈ ℤ The function will be decreasing in the interval where f ' ( x ) < 0 So we need to Solve the inequality 1 4 cos ( x 4 ) < 0 4 ⋅ 1 4 cos ( x 4 ) < 4 ⋅ 0 cos ( x 4 ) < 0 ( 8 π n + 2 π , 8 π n + 6 π ) , n ∈ ℤ Calculation (Con

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 3, Problem 6SC

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Chapter 3 Solutions

Calculus

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where the function is increasing or decreasing.

Solution Summary: The author explains how to find the first and second derivatives of the given function.

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Chapter 3 Solutions