Explanation Given: The given function is: f ( x ) = − 2 x 3 − 2 x 2 + 2 x − 1 and the interval is: [ − 3 , 1 ] Approach: Finding absolute extrema: To find the absolute extrema for a function f continuous on a closed interval [ a , b ] : 1. Find all critical numbers for f in ( a , b ) . 2. Evaluate f for all critical numbers in ( a , b ) . 3. Evaluate f for the endpoints a and b of the interval [ a , b ] . 4. The largest value found in Step 2 or 3 is the absolute maximum for f on [ a , b ] , and the smallest value found is the absolute minimum for f on [ a , b ] . Calculation: Consider the given function, f ( x ) = − 2 x 3 − 2 x 2 + 2 x − 1 Differentiate the function with respect to x : f ′ ( x ) = − 6 x 2 − 4 x + 2 Find all critical points for f : f ′ ( x ) = 0 − 6 x 2 − 4 x + 2 = 0 3 x 2 + 2 x − 1 = 0 − 2 ( x + 1 ) ( 3 x − 1 ) = 0 Further simplify as: ( x + 1 ) ( 3 x − 1 ) = 0 x = − 1 or 1 3 The value of the function at the critical point and as well as end point: At x = − 3 :

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Chapter 6.CR, Problem 12CR

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The absolute extrema and all values of x for the given function f(x)=−2x3−2x2+2x−1 and interval [−3,1].

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Chapter 6.CR, Problem 11CR

Chapter 6.CR, Problem 13CR

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Chapter 6 Solutions

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

Ch. 6.1 - YOUR TURN Find the absolute extrema of the...Ch. 6.1 - Prob. 2YT Ch. 6.1 - EXERCISES Find the locations of any absolute...Ch. 6.1 - EXERCISES Find the locations of any absolute...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - EXERCISES Find the locations of any absolute...Ch. 6.1 - EXERCISES Find the locations of any absolute...Ch. 6.1 - Prob. 8E

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The absolute extrema and all values of x for the given function f ( x ) = − 2 x 3 − 2 x 2 + 2 x − 1 and interval [ − 3 , 1 ] .

Solution Summary: The author explains how to find the absolute extrema and all values of x for a function f continuous on an interval left[a,bright]

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To find:

The absolute extrema and all values of x for the given function f(x)=−2x3−2x2+2x−1 and interval [−3,1].

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Chapter 6 Solutions

The absolute extrema and all values of x for the given function f ( x ) = − 2 x 3 − 2 x 2 + 2 x − 1 and interval [ − 3 , 1 ] .

Solution Summary: The author explains how to find the absolute extrema and all values of x for a function f continuous on an interval left[a,bright]

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To find: The absolute extrema and all values of x for the given function f(x)=−2x3−2x2+2x−1 and interval [−3,1].

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Chapter 6 Solutions

To find:

The absolute extrema and all values of x for the given function f(x)=−2x3−2x2+2x−1 and interval [−3,1].