Explanation Given: The function of the graph is given as: f ( x ) = − 2 x 3 − 9 x 2 + 60 x − 8 Graph: The steps to be followed in TI-83 are as follows: Step 1: In the TI-83 graphical calculator, press the Y = key. Step 2: Enter the equation of f ( x ) as − 2 X ^ 3 − 9 X ^ 2 + 108 X − 10 in Y 1 . Step 3: Press the TRACE and Y key. Step 4: Go to WINDOW and set X min = − 1 000, X max =1000, Y min = − 10, Y max =10 . Step 5: Click the TRACE key to obtain the graph. The graph is obtained as: For y -intercept: f ( 0 ) = − 8 For x- intercept: f ( x ) = 0 − 2 x 3 − 9 x 2 + 60 x − 8 = 0 x = 3.57 , − 8.212 , 0.136 The formula used to find 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 ( − 2 x 3 − 9 x 2 + 60 x − 8 ) = d d x ( − 2 x 3 ) + d d x ( − 9 x 2 ) + d d x ( 60 x ) − d d x ( 8 ) = − 2 ⋅ 3 x 2 − 9 ⋅ 2 x + 60 − 0 = − 6 x 2 − 18 x + 60 Now put f ′ ( x ) = 0 to find critical points: − 6 x 2 − 18 x + 60 = 0 x 2 + 3 x − 10 = 0 x 2 + 5 x − 2 x − 10 = 0 x ( x + 5 ) − 2 ( x + 5 ) = 0 Upon further calculations: ( x − 2 ) ( x + 5 ) = 0 x = 2 , − 5 Now find the second-order derivative of the function: f ″ ( x ) = d d x ( − 6 x 2 − 18 x + 60 ) = d d x ( − 6 x 2 ) + d d x ( − 18 x ) + d d x ( 60 ) = − 6 ⋅ 2 x − 18 + 0 = − 12 x − 18 To get the point of local extrema put x = 2 , − 5 and find value of f ″ ( x ) at critical points: f ″ ( 2 ) = − 12 ( 2 ) − 18 = − 24 − 18 = − 42 And: f ″ ( − 5 ) = − 12 ( − 5 ) − 18 = 60 − 18 = 42 The second-order derivative of function at x = 2 is negative, so it is the point of local maxima and at x = − 5 is positive hence the point is a local minimum

12th Edition

ISBN: 9780134773285

Author: MULLINS

Publisher: YUZU

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Chapter 12.6, Problem 6E

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To graph: The function f(x)=−2x3−9x2+60x−8 and locate the point of inflection and local extrema.

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Chapter 12.6, Problem 5E

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Chapter 12 Solutions

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To graph: The function f ( x ) = − 2 x 3 − 9 x 2 + 60 x − 8 and locate the point of inflection and local extrema.

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To graph: The function f(x)=−2x3−9x2+60x−8 and locate the point of inflection and local extrema.

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