Explanation Given information :. f ( x ) = x 2 ( x − 1 ) ( x + 1 ) Formula Used: Power rule of derivative: If f ( x ) = k x n then f ' ( x ) = k ⋅ n x n − 1 Algebraic identity: ( a − b ) ( a + b ) = a 2 − b 2 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 Rewrite the given function by multiplying first outer inner last terms of ( x − 1 ) ( x + 1 ) and then distributing x 2 f ( x ) = x 2 ( x − 1 ) ( x + 1 ) = x 2 ( x 2 − 1 ) = x 4 − x 2 Use power rule to find the derivative of the function f ( x ) = x 4 − x 2 f ' ( x ) = d ( x 4 ) / d x − d ( x 2 ) / d x = 4 x 4 − 1 − 2 x 2 − 1 = 4 x 3 − 2 x Factor the first derivative by factoring out the greatest common factor of f ' ( x ) = 4 x 3 − 2 x f ' ( x ) = 2 x ( 2 x 2 − 1 ) The critical number is the value of x that makes the first derivative equal to zero or undefined. 2 x = 0 be the 1st equation 2 x 2 − 1 = 0 be the 2nd equation Solving the 1 s t equation 2 x = 0 by • Dividing 2 on both sides 2 x / 2 = 0 / 2 x = 0 Solving the 2 n d equation 2 x 2 − 1 = 0 by • Adding 1 on both sides • Dividing 2 on both sides • Taking square root on both sides 2 x 2 − 1 + 1 = 0 + 1 2 x 2 = 1 2 x 2 / 2 = 1 / 2 x 2 = 0.5 x 2 = 0.5 x = ± 0.5 List the values of x {the critical numbers} that makes the first derivative of given function as zero x = − 0.5 ; x = 0 ; x = 0.5 List all the intervals where the function will be increasing or decreasing using the critical number. ( − ∞ , − 0.5 ) ; ( − 0.5 , 0 ) ; ( 0 , 0.5 ) ( 0.5 , ∞ ) Pick a test point from each interval and check the sign of f ' ( x ) = 2 x ( 2 x 2 − 1 ) Substituting the test point x = − 1 ∈ ( − ∞ , − 0.5 ) f ' ( − 1 ) = 2 ( − 1 ) ( 2 ( − 1 ) 2 − 1 ) = − 2 < 0 Substituting the test point x = − 0.1 ∈ ( − 0.5 , 0 ) , f ' ( − 0.1 ) = 2 ( − 0.1 ) ( 2 ( − 0.1 ) 2 − 1 ) = 0

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 3.2, Problem 40S

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To find : the local extrema of the function.

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Chapter 3.2, Problem 39S

Chapter 3.2, Problem 41S

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Chapter 3 Solutions

Calculus

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the local extrema of the function.

Solution Summary: The author explains how to find the local extrema of the function. Critical numbers are values that make the first derivative a zero or undefined.

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Chapter 3 Solutions