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 )

Pearson eText Calculus: Early Transcendentals -- Instant Access (Pearson+)

3rd Edition

ISBN: 9780136880677

Author: William Briggs

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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

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Pearson eText Calculus: Early Transcendentals -- Instant Access (Pearson+)

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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