Explanation Given information :. f ( x ) = cos ( π x ) 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: Step 1: Find the derivative. Step 2: Critical numbers are values of x that makes the first derivative a zero or undefined. Set up equation to find the critical numbers ‘c’. Step 3: Write all the intervals where the function will be increasing or decreasing using the critical number. Step 4: Pick a test point ‘ x 0 ’ from each interval, substitute it in the first derivative; If f ' ( x 0 ) < 0 , then function is decreasing in that interval& If f ' ( x 0 ) > 0 , then function is increasing in that interval. Step 5: Based on the below cases, identify whether the critical number that we found is local maximum or local minimum. If to the left of critical number function is decreasing and to the right function is increasing, then it means at c we have local minimum given by ( c , f ( c ) ) If to the left of critical number function is increasing and to the right function decreasing, then it means at c we have local maximum. given by ( c , f ( c ) ) Calculation: Description Steps Find the derivative of the function f ( x ) = cos ( π x ) f ' ( x ) = − sin ( π x ) ⋅ d d x ( π x ) = − sin ( π x ) ⋅ ( π ) = − π sin ( π x ) The critical number is the value of x that makes the first derivative f ' ( x ) = − π sin ( π x ) equal to zero or undefined

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 3.2, Problem 46S

To determine

To find : the local extrema of the function.

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Chapter 3.2, Problem 45S

Chapter 3.2, Problem 47S

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