Explanation Given: The function is f ( x ) = − 2 3 x 3 + 9 2 x 2 + 5 x + 1 . Graph: Step 1: Find the y intercept of the function by putting x = 0 , f ( x ) = − 2 3 x 3 + 9 2 x 2 + 5 x + 1 f ( 0 ) = 1 Step 2: Find the x- intercept by equating the equation to 0, f ( x ) = − 2 3 x 3 + 9 2 x 2 + 5 x + 1 0 = − 2 3 x 3 + 9 2 x 2 + 5 x + 1 Since the equation is polynomial equation, it is not easy to find x by hand and it will not have any asymptotes. Step 3: Find the first and second derivative of f ( x ) = − 2 3 x 3 + 9 2 x 2 + 5 x + 1 as, f ' ( x ) = − 2 x 2 + 9 x + 5 f " ( x ) = − 4 x + 9 Step 4: Find the critical numbers by equating f ' ( x ) = 0 as, f ' ( x ) = − 2 x 2 + 9 x + 5 0 = − 2 x 2 + 9 x + 5 2 x 2 − 9 x − 5 = 0 ( 2 x + 1 ) ( x − 5 ) = 0 On solving, x = 5 , − 1 2 Find the value of second derivative using the critical numbers that is, f " ( x ) = − 4 x + 9 f " ( 5 ) = − 4 ⋅ 5 + 9 = − 11 f " ( 5 ) < 0 And, f " ( x ) = − 4 x + 9 f " ( − 1 2 ) = − 4 ⋅ ( − 1 2 ) + 9 = 7 f " ( − 1 2 ) > 0 Local maxima is at x = − 1 2 and local minima is at x = 5 . Step 5: Determine the intervals on which the graph is increasing the decreasing using critical numbers by evaluating f ' ( x ) . The function f ( x ) is increasing when f ' ( x ) > 0 and decreasing f ' ( x ) < 0

Mathematics with Applications, Books a la Carte, and MyLab Math with Pearson eText - Title-Specific Access Card Package (12th Edition)

12th Edition

ISBN: 9780134862651

Author: Lial

Publisher: PEARSON

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Chapter 12, Problem 24RE

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To graph: The function f(x)=−23x3+92x2+5x+1 and list the local extremums and point of inflections, intervals on which the function is increasing and decreasing and intervals in which the function is concave upward and concave downward.

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Chapter 12, Problem 23RE

Chapter 12, Problem 25RE

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Chapter 12 Solutions

Mathematics with Applications, Books a la Carte, and MyLab Math with Pearson eText - Title-Specific Access Card Package (12th Edition)

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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To graph: The function f ( x ) = − 2 3 x 3 + 9 2 x 2 + 5 x + 1 and list the local extremums and point of inflections, intervals on which the function is increasing and decreasing and intervals in which the function is concave upward and concave downward.

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To graph: The function f(x)=−23x3+92x2+5x+1 and list the local extremums and point of inflections, intervals on which the function is increasing and decreasing and intervals in which the function is concave upward and concave downward.

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Chapter 12 Solutions