(a) Explanation Given information : f ( x ) = 2 x 3 − 3 x 2 − 12 x on the interval [-3,3] Formula Used: Power rule of derivative: If $f(x)=k x^{n}$ then $f^{\prime}(x)=k \cdot n x^{n-1}$ Step 1: Find the first derivative of the function. 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 and the given end points 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. 5: Based on the below cases, identify whether the critical number that we found is local Step 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. Description Steps Find the derivative of the function f ( x ) = 2 x 3 − 3 x 2 − 12 x The critical number is the value of x that makes the first derivative equal to zero or undefined. f ′ ( x ) = 6 ( x − 2 ) ( x + 1 ) 6 ( x − 2 ) = 0 x + 1 = 0 Solving the equation − x = 0 ➢Divide by 6 on both sides ➢Add 2 on both sides 6 ( x − 2 ) 6 = 0 6 x − 2 = 0 x − 2 + 2 = 0 + 2 x = 2 Solving the equation 1 + x = 0 ➢Subtract 1 on both sides x + 1 − 1 = 0 − 1 x = − 1 List the values of x {the critical numbers} that makes the first derivative of given function as zero x = − 1 ; List all the intervals where the function will be increasing or decreasing using the critical number and the end points. [ − 3 , − 1 ) ; ( − 1 , 2 ) ; ( 2 , 3 ] Pick a test point from each interval and check the sign of first derivative f ′ ( x ) = 6 ( x − 2 ) ( x + 1 ) Substituting x = − 2 ∈ [ − 3 , − 1 ) Substituting x = 0 ∈ ( − 1 , 2 ) Substituting x = 2.5 ∈ ( 2 , 3 ] Sign chart of the first derivative (b) To determine To find : the local extrema and global extrema of the function in the given interval (c) To determine To find : the local extrema and global extrema of the function in the given interval (d) To determine To find : the local extrema and global extrema of the function in the given interval

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 3.4, Problem 11S

(a)

To determine

To find : the local extrema and global extrema of the function in the given interval

(b)

To determine

To find : the local extrema and global extrema of the function in the given interval

(c)

To determine

To find : the local extrema and global extrema of the function in the given interval

(d)

To determine

To find : the local extrema and global extrema of the function in the given interval

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Chapter 3.4, Problem 10C

Chapter 3.4, Problem 12S

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Chapter 3 Solutions

Calculus

Ch. 3.1 - Prob. 1TB Ch. 3.1 - Prob. 0C Ch. 3.1 - Prob. 1C Ch. 3.1 - Prob. 2C Ch. 3.1 - Prob. 3C Ch. 3.1 - Prob. 4C Ch. 3.1 - Prob. 5C Ch. 3.1 - Prob. 6C Ch. 3.1 - Prob. 7C Ch. 3.1 - Prob. 8C

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the local extrema and global extrema of the function in the given interval

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To find : the local extrema and global extrema of the function in the given interval

To find : the local extrema and global extrema of the function in the given interval

To find : the local extrema and global extrema of the function in the given interval

To find : the local extrema and global extrema of the function in the given interval

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Chapter 3 Solutions