Explanation Given: The cubic function is f ( x ) = x 3 + a x 2 + b x + c and the domain of the cubic function is ( − ∞ , ∞ ) . Concept used: Assume f be a differentiable function on an open interval ( a , b ) then f is increasing on ( a , b ) if f ′ ( x ) > 0 for x ∈ ( a , b ) and f is decreasing on ( a , b ) if f ′ ( x ) < 0 for x ∈ ( a , b ) . Calculation: The cubic function is represented below. f ( x ) = x 3 + a x 2 + b x + c ...... (1) Differentiate equation (1) as shown below: f ′ ( x ) = d ( x 3 + a x 2 + b x + c ) d x = 3 x 2 + 2 a x + b By completing the square technique, rewrite the above equation as shown below: f ′ ( x ) = 3 x 2 + 2 a x + b = 3 x 2 + 2 a x + a 2 3 − a 2 3 + b = 3 ( x 2 + 2 3 a x + a 2 3 2 ) − a 2 3 + b = 3 ( x + a 3 ) 2 + b − a 2 3 Label the above equation as equation (2)

CALCULUS EARLY TRANSCENDENTALS W/ACHIEVE

4th Edition

ISBN: 9781319057664

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 4.3, Problem 72E

To determine

To show:

The cubic function f(x)=x3+ax2+bx+c is increasing on (−∞,∞) if b>a23.

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Chapter 4.3, Problem 71E

Chapter 4.3, Problem 73E

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Chapter 4 Solutions

CALCULUS EARLY TRANSCENDENTALS W/ACHIEVE

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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The cubic function f ( x ) = x 3 + a x 2 + b x + c is increasing on ( − ∞ , ∞ ) if b > a 2 3 .

Solution Summary: The author illustrates how the cubic function f(x) is a differentiable function on an open interval.

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The cubic function f(x)=x3+ax2+bx+c is increasing on (−∞,∞) if b>a23.

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Chapter 4 Solutions

The cubic function f ( x ) = x 3 + a x 2 + b x + c is increasing on ( − ∞ , ∞ ) if b &gt; a 2 3 .

Solution Summary: The author illustrates how the cubic function f(x) is a differentiable function on an open interval.

Concept explainers

Videos

To show: The cubic function f(x)=x3+ax2+bx+c is increasing on (−∞,∞) if b>a23.

Want to see the full answer?

Chapter 4 Solutions

The cubic function f ( x ) = x 3 + a x 2 + b x + c is increasing on ( − ∞ , ∞ ) if b > a 2 3 .

To show:

The cubic function f(x)=x3+ax2+bx+c is increasing on (−∞,∞) if b>a23.