Explanation Result used: If f ′ ( x ) > 0 on an interval, then f is increasing on that interval. If f ′ ( x ) < 0 on an interval, then f is decreasing on that interval. The points where the first derivative is 0 is called the critical points. The points where the second derivative is 0 is called the inflection points. The graph of y = f ( x ) is concave upward on those intervals where f ″ ( x ) > 0 . The graph of y = f ( x ) is concave downward on those intervals where f ″ ( x ) < 0 . Calculation: Consider the function f ( x ) = e x ( x 2 − x ) . Clearly the domain of the function is the set of all real numbers. Compute the derivative of f ( x ) = e x ( x 2 − x ) . f ′ ( x ) = d d x ( e x ( x 2 − x ) ) = e x d d x ( x 2 − x ) + ( x 2 − x ) d d x ( e x ) [ By product rule ] = e x ( 2 x − 1 ) + ( x 2 − x ) e x = e x ( x + x 2 − 1 ) Set f ′ ( x ) = 0 . e x ( x + x 2 − 1 ) = 0 ( x + x 2 − 1 ) = 0 [ ∵ e x ≠ 0 ] x = − 1 ± 1 + 4 2 x = − 1 ± 5 2 The nature of f ′ ( x ) for various values of x is tabulated as follows. Interval Nature of f ′ ( x ) ( − ∞ , − 1 − 5 2 ) f ′ ( x ) > 0 ( − 1 − 5 2 , − 1 + 5 2 ) f ′ ( x ) < 0 ( − 1 + 5 2 , ∞ ) f ′ ( x ) > 0 Hence, f is increasing on the intervals ( − ∞ , − 1 − 5 2 ) , ( − 1 + 5 2 , ∞ ) and decreasing on the interval ( − 1 − 5 2 , − 1 + 5 2 )

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 7, Problem 27RE

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To graph: The function f(x)=ex(x2−x) and identify the local extreme points, inflection points, concavity, and end behaviour.

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Chapter 7, Problem 26RE

Chapter 7, Problem 28RE

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Sketch the graph of f.

PART ONE: f(x) = ab* ➤ In this part, you are going to explore the behavior of the function using various values of the parameters a & b. 1. When will the function have the following properties: x → ∞o, y → 0 Describe the parameters a & b that leads to the property above. Follow up: Print three functions of this type on one Cartesian Plane. Make sure to correctly label the functions. 2. When will the function have the following properties: x →∞o, y co Describe the parameters a & b that leads to the property above. Follow up: Print three functions of this type on one Cartesian Plane. Make sure to correctly label the functions. 3. The first two questions only covered the "two types". In fact, there are "four types" of f(x) = ab* in terms of "trend". You are tasked to explore the other two. TASK: Upon exploration, make a detailed summary of the relationships of the "four types".

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 7.1 - What is the domain of ln |x|?Ch. 7.1 - Quick Check 2 Simplify e ln 2x, ln (e2x), e2 ln x,...Ch. 7.1 - Prob. 3QC Ch. 7.1 - Prob. 4QC Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Evaluate 4xdx.Ch. 7.1 - Prob. 4E Ch. 7.1 - Express 3x, x, and xsin x using the base e.Ch. 7.1 - Prob. 6E

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To graph: The function f ( x ) = e x ( x 2 − x ) and identify the local extreme points, inflection points, concavity, and end behaviour.

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To graph: The function f(x)=ex(x2−x) and identify the local extreme points, inflection points, concavity, and end behaviour.

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