To determine To graph: The function f ( x ) = 2 x 3 + 7 2 x 2 − 5 x + 3 and locate the point of inflection and local extrema. Explanation Given: The function is given as: f ( x ) = 2 x 3 + 7 2 x 2 − 5 x + 3 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 + ( 7 ÷ 2 ) X ^ 2 − 5 X + 3 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: The x - and y- intercepts are obtained as: For y -intercept: f ( 0 ) = 3 For x- intercept: f ( x ) = 0 2 x 3 + 7 2 x 2 − 5 x + 3 = 0 x = − 2.82 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 + 7 2 x 2 − 5 x + 3 ) = d d x ( 2 x 3 ) + d d x ( 7 2 x 2 ) − d d x ( 5 x ) + d d x ( 3 ) = 2 ⋅ 3 x 2 + 7 2 ⋅ 2 x − 5 + 0 = 6 x 2 + 7 x − 5 Now put f ′ ( x ) = 0 to find critical points: 6 x 2 + 7 x − 5 = 0 6 x 2 + 10 x − 3 x − 5 = 0 2 x ( 3 x + 5 ) − ( 3 x + 5 ) = 0 ( 2 x − 1 ) ( 3 x + 5 ) = 0 Upon further calculation: x = 1 2 , − 5 3 Now find the second-order derivative of the function: f ″ ( x ) = d d x ( 6 x 2 + 7 x − 5 ) = d d x ( 6 x 2 ) + d d x ( 7 x ) − d d x ( 5 ) = 6 ⋅ 2 x + 7 + 0 = 12 x + 7 To get the point of local extrema put x = 1 2 , − 5 3 to find the value of f ″ ( x ) at critical points: f ″ ( 1 2 ) = 12 ( 1 2 ) + 7 = 6 + 7 = 13 And: f ″ ( − 5 3 ) = 12 ( − 5 3 ) + 7 = − 20 + 7 = − 13 The second-order derivative of function at x = − 5 3 is negative, so it is the point of local maxima and at x = 1 2 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 7E

To determine

To graph: The function f(x)=2x3+72x2−5x+3 and locate the point of inflection and local extrema.

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Chapter 12.6, Problem 6E

Chapter 12.6, Problem 8E

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