Explanation Given: The function is f ( x ) = 3 x ⋅ e − x . Formula used: According to the first derivative test, if c is the only critical number of the given function for an interval [ a , b ] such that a < c < b . Consider, f is differential for every value of x in the interval except at x = c . 1. For f ′ ( a ) > 0 and f ′ ( b ) < 0 , there will be a local maxima at c . 2. For f ′ ( a ) < 0 and f ′ ( b ) > 0 , there will be a local minima at c . 3. There will be no local extrema when f ′ ( a ) and f ′ ( b ) are either both positive or both negative. And, Product rule, d d x [ f ( x ) g ( x ) ] = g ( x ) d d x [ f ( x ) ] + f ( x ) d d x [ g ( x ) ] Calculation: Solve the given function: f ( x ) = 3 x ⋅ e − x Differentiate the above function: f ′ ( x ) = − 3 x ⋅ e − x + 3 e − x = 3 e − x ( − x + 1 ) Substitute 0 for f ′ ( x ) : 3 e − x ( − x + 1 ) = 0 Since 3 e − x can never be 0

MyLab Math with Pearson eText -- Standalone Access Card -- for Mathematics with Applications in the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134856551

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

Publisher: PEARSON

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Chapter 12, Problem 15RE

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To calculate: The value of local maximum and minimum and their locations of the function f(x)=3x⋅e−x.

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Chapter 12, Problem 14RE

Chapter 12, Problem 16RE

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What is a solution to a differential equation? We said that a differential equation is an equation that describes the derivative, or derivatives, of a function that is unknown to us. By a solution to a differential equation, we mean simply a function that satisfies this description. 2. Here is a differential equation which describes an unknown position function s(t): ds dt 318 4t+1, ds (a) To check that s(t) = 2t2 + t is a solution to this differential equation, calculate you really do get 4t +1. and check that dt' (b) Is s(t) = 2t2 +++ 4 also a solution to this differential equation? (c) Is s(t)=2t2 + 3t also a solution to this differential equation? ds 1 dt (d) To find all possible solutions, start with the differential equation = 4t + 1, then move dt to the right side of the equation by multiplying, and then integrate both sides. What do you get? (e) Does this differential equation have a unique solution, or an infinite family of solutions?

Chapter 12 Solutions

MyLab Math with Pearson eText -- Standalone Access Card -- for 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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The value of local maximum and minimum and their locations of the function f ( x ) = 3 x ⋅ e − x .

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To calculate: The value of local maximum and minimum and their locations of the function f(x)=3x⋅e−x.

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Chapter 12 Solutions