Explanation Given: The function is given as: f ( x ) = x e − x Graph: To plot the graph in the graphing calculator, follow the below-mentioned steps: Step 1: In the TI-83 graphical calculator, press the Y = key. Step 2: Enter the equation of f ( x ) as X e ^ ( − X ) in Y 1 . Step 3: Press the TRACE and Y key. Step 4: Go to WINDOW and set X min = − 1 , X max = 4 , Y min = − 4 , Y max = 1 . Step 5: Click on the TRACE key to obtain the graph. The graph is obtained as follows: The x - and y -intercepts are obtained as: The y -intercept is: f ( 0 ) = 0 ⋅ e − 0 f ( 0 ) = 0 To find the x -intercepts, solve f ( x ) = 0 . x e − x = 0 x e = 0 x = 0 The x- intercept is x = 0 . Power rule of differentiation states that if f ( x ) = x n for any number n, then f ′ ( x ) = n x n − 1 . Product rule of differentiation states that if: f ( x ) = u ( x ) ⋅ v ( x ) Then, f ′ ( x ) is given by: f ′ ( x ) = u ′ ( x ) ⋅ v ( x ) + u ( x ) ⋅ v ′ ( x ) To find all the critical points, calculate the first-order derivative of the given function: f ′ ( x ) = d f ( x ) d x = d ( x ) d x ⋅ e − x + d ( e − x ) d x ⋅ x = ( 1 ) ⋅ e − x − e − x ⋅ x = e − x ( 1 − x ) Now put f ′ ( x ) = 0 to find critical points: e − x ( 1 − x ) = 0 x = 1 Now find the second-order derivative of the function: f ″ ( x ) = d f ′ ( x ) d x Use the product rule: f ″ ( x ) = d d x ( e − x ( 1 − x ) ) = d ( e − x ) d x ⋅ ( 1 − x ) + d ( 1 − x ) d x ⋅ ( e − x ) = e − x ( − 1 ) ⋅ ( 1 − x ) + ( 0 − 1 ) ⋅ e − x = e − x ( − 1 ) ⋅ ( 2 − x ) Assume y = f ( x ) , To find a point of inflection, put f ″ ( x ) = 0

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 21E

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To graph: The function f(x)=xe−x and locate the point of inflection and local extrema.

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Chapter 12.6, Problem 20E

Chapter 12.6, Problem 22E

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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 ) = x e − x and locate the point of inflection and local extrema.

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Chapter 12 Solutions