Explanation Given: The function is f ( x ) = 2 x 3 − 3 x 2 − 12 x + 2 . Formula used: According to the first derivative test, if c is the only critical number of the given function for an interval [ a , b ] such that a < c < b . Consider, f is differential for every value of x in the interval except at x = c . 1. For f ′ ( a ) > 0 and f ′ ( b ) < 0 , there will be a local maxima at c . 2. For f ′ ( a ) < 0 and f ′ ( b ) > 0 , there will be a local minima at c . 3. There will be no local extrema when f ′ ( a ) and f ′ ( b ) are either both positive or both negative Calculation: Solve the given function: f ( x ) = 2 x 3 − 3 x 2 − 12 x + 2 Differentiate the above function”: f ′ ( x ) = 6 x 2 − 6 x − 12 = 6 x 2 − 12 x + 6 x − 12 = 6 x ( x − 2 ) + 6 ( x − 2 ) = ( 6 x + 6 ) ( x − 2 ) Substitute 0 for f ′ ( x ) . ( 6 x + 6 ) ( x − 2 ) = 0 One critical number is: 6 x + 6 = 0 x = − 6 6 = − 1 Other critical number is: x − 2 = 0 x = 2 Thus, − 1 and 2 are the critical numbers for the function. Use first derivative test separately on − 1 and 2 to determine the local extrema of the given function. Let choose a point − 2 less than − 1 and 0 greater than − 1 . Calculate the value of f ′ ( − 2 ) . f ′ ( − 2 ) = ( 6 ( − 2 ) + 6 ) ( − 2 − 2 ) = ( − 12 + 6 ) ( − 4 ) = ( − 6 ) ( − 4 ) = 24 Calculate the value of f ′ ( 0 )

Mathematics with Applications and Mylab Math with Pearson EText -- Title-Specific Access Card Package

12th Edition

ISBN: 9780134862668

Author: Lial, Margaret L.

Publisher: Pearson Education Canada

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Chapter 12, Problem 12RE

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Chapter 12, Problem 11RE

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Chapter 12 Solutions

Mathematics with Applications and Mylab Math with Pearson EText -- Title-Specific Access Card Package

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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The value of local maxima and minima and their locations of the function f ( x ) = 2 x 3 − 3 x 2 − 12 x + 2 .

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