Explanation Given: The function is given as: f ( x ) = 3 x 3 − 3 x 2 + 1 Graph: To trace the graph in TI-83, follow the below mentioned steps: Step 1. In the TI-83 graphical calculator, press the Y= button. Step 2. Enter the equation of f ( x ) as 3 X ^ 3 − 3 X ^ 2 + 1 in Y 1 . Step 3. Press the TRACE and Y button. Step 4. Go to WINDOW and set: X min = − 10 X max = 10 Y min = − 5 Y max = 5 Step 5. Click the TRACE button to obtain the graph. The obtained graph is as follows: For y intercept, f ( 0 ) = 1 . For x intercept, f ( x ) = 0 3 x 3 − 3 x 2 + 1 = 0 x = − 0.475 The formula used to find the first order and second order derivatives are: f ′ ( x ) = d d x ( f ( x ) ) f ″ ( x ) = d d x ( f ′ ( x ) ) Calculate the first order derivative of the given function: f ′ ( x ) = d d x ( 3 x 3 − 3 x 2 + 1 ) = d d x ( 3 x 3 ) − d d x ( 3 x 2 ) + d d x ( 1 ) = 3 ⋅ 3 x 2 − 3 ⋅ 2 x + 0 = 9 x 2 − 6 x Now, put f ′ ( x ) = 0 to find the critical points: 9 x 2 − 6 x = 0 3 x ( 3 x − 2 ) = 0 x = 0 , 2 3 Now find the second order derivative of the function as: f ″ ( x ) = d d x ( 9 x 2 − 6 x ) = d d x ( 9 x 2 ) − d d x ( 6 x ) = 9 ⋅ 2 x − 6 = 18 x − 6 To get the point of local extrema put find value of f ″ ( x ) at critical points: f ″ ( 0 ) = 18 ( 0 ) − 6 = 0 − 6 = − 6 f ″ ( 2 3 ) = 18 ( 2 3 ) − 6 = 12 − 6 = 6 The second order derivative of function at x = 0 is negative

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.6, Problem 3E

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Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (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 ) = 3 x 3 − 3 x 2 + 1 .

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