Explanation Given information : f ( x ) = 1 1 + x Formula Used: Power rule of derivative: If f ( x ) = k x n then f ' ( x ) = k ⋅ n x n − 1 Quotient Rule : d d x ( f g ) = g ⋅ f ' − f ⋅ g ' g 2 Chain rule: ( f ( g ( x ) ) ) ' = f ' ( g ( x ) ) ⋅ g ' ( x ) Concept Involved: Step 1: The roots of the function are the values of x that makes the function equal to zero. Step 2: Find the first derivative of the function. Step 3: 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 4: Write all the intervals where the function will be increasing or decreasing using the critical number. Step 5: 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 6: 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 ) ) Global extrema are the largest and smallest values that a function takes on over its entire domain Step 7: Find the second derivative of the function. Step 8: Find the values of x that makes the f ' ' ( x ) = 0 . Step 9: Write all the intervals where the function will be concave up or concave down using the values of x that makes the equation f ' ' ( x ) = 0 TRUE. Step 10: Pick a test point ‘ x 1 ’ from each interval, substitute it in the first derivative; If f ' ' ( x 1 ) < 0 , then function is concave down in that interval & If f ' ' ( x 1 ) > 0 , then function is concave up in that interval. Step 11: Inflection points are the values of x where the function changes concavity. Calculation: Description Steps Find the zeros of the function that makes f ( x ) = 0 . For a rational function we need to set the numerator to zero There is no x value that makes the given function to zero List the values of x that makes the function as zero f ( x ) = 0 N o R o o t s Rewrite the given function as a negative exponent f ( x ) = ( 1 + x 1 2 ) − 1 Find the derivative of the function f ( x ) = 1 1 + x f ' ( x ) = − 1 ( 1 + x 1 2 ) − 2 ⋅ d d x ( 1 + x 1 2 ) = − 1 ( 1 + x 1 2 ) − 2 ⋅ ( 1 2 x − 1 2 ) = − 1 ( 1 + x ) 2 ⋅ 1 2 x = − 1 2 x ( 1 + x ) 2 Calculation (Continued): Description Steps The critical number is the value of x that makes the first derivative f ' ( x ) = − 1 2 x ( 1 + x ) 2 equal to zero or undefined. Setting up numerator and denominator to zero 2 x = 0 be the 1 s t equation ( 1 + x ) 2 = 0 be the 2 n d equation Solving the equation 2 x = 0 by Dividing by -1 on both sides Squaring on both sides 2 x / 2 = 0 / 2 x = 0 ( x ) 2 = ( 0 ) 2 x = 0 Solving the equation ( 1 + x ) 2 = 0 by There is no real value of xthat makes the equation TRUE List the critical number that makes first derivative as undefined x = 0 List the intervals using the critical number where the function will be increasing or decreasing Note: the domain will be all the values of x that are greater than or equal to zero because negative inside the square root will make the function undefined ( 0 , ∞ ) Pick a test point from each interval and check the sign of f ' ( x ) = − 1 2 x ( 1 + x ) 2 Substituting the test point x = 1 ∈ ( 0 , ∞ ) , f ' ( 0 ) = − 1 / 2 1 ( 1 + 1 ) 2 = − 0.125 < 0 Analyzing the left and right of critical number x = 1 Since the function is not defined to the left of 0. There is no local minimum or local maximum to this function Rewrite the first derivative using the rational exponent and negative exponent rule f ' ( x ) = − 1 2 x ( 1 + x ) 2 = − 1 2 x 1 2 ( 1 + x ) 2 = − 0.5 x − 1 2 ( 1 + x 1 2 ) − 2 Calculation (Continued): Description Steps Find the second derivative of the function f ' ' ( x ) = d ( f ' ( x ) ) / d x = d ( − 0

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 3, Problem 12SC

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Chapter 3, Problem 11SC

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Chapter 3 Solutions

Calculus

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the roots, local extrema and inflection points of given function.

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Chapter 3 Solutions