To determine To graph: The function f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 and locate the point of inflection and local extrema. Explanation Given: The function is given as: f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 Graph: The steps to be followed to plot the curve in the graphical calculator 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 follows: For y -intercept: f ( 0 ) = − 10 For x- intercept: f ( x ) = 0 − 2 x 3 − 9 x 2 + 108 x − 10 = 0 x = 5.37 , − 9.96 , 0.093 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 ( − 2 x 3 − 9 x 2 + 108 x − 10 ) = d d x ( − 2 x 3 ) + d d x ( − 9 x 2 ) + d d x ( 108 x ) − d d x ( 10 ) = − 2 ⋅ 3 x 2 − 9 ⋅ 2 x + 108 − 0 = − 6 x 2 − 18 x + 108 Now put f ′ ( x ) = 0 to find critical points: − 6 x 2 − 18 x + 108 = 0 x 2 + 3 x − 18 = 0 x 2 + 6 x − 3 x − 18 = 0 x ( x + 6 ) − 3 ( x + 6 ) = 0 Upon further calculation: ( x − 3 ) ( x + 6 ) = 0 x = 3 , − 6 Now find the second-order derivative of the function: f ″ ( x ) = d d x ( − 6 x 2 − 18 x + 108 ) = d d x ( − 6 x 2 ) + d d x ( − 18 x ) + d d x ( 108 ) = − 6 ⋅ 2 x − 18 + 0 = − 12 x − 18 To get the point of local extrema put x = 3 , − 6 and find value of f ″ ( x ) at critical points: f ″ ( 3 ) = − 12 ( 3 ) − 18 = − 36 − 18 = − 54 And: f ″ ( − 6 ) = − 12 ( − 6 ) − 18 = 72 − 18 = 54 The second-order derivative of function at x = 3 is negative, so it is the point of local maxima and at x = − 6 is positive hence the point is a local minimum

12th Edition

ISBN: 9780135335215

Author: Lial

Publisher: PEARSON

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Chapter 12.6, Problem 5E

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To graph: The function f(x)=−2x3−9x2+108x−10 and locate the point of inflection and local extrema.

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Chapter 12.6, Problem 4E

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