Explanation Given information: The function is f ( x ) = ( x 2 + 1 ) ( 2 − x ) and the tangent line is horizontal. Formula used : The product rule of differentiation is, f ′ ( x ) = d d x [ f ( x ) ⋅ g ( x ) ] = f ( x ) ⋅ d d x g ( x ) + g ( x ) ⋅ d d x f ( x ) Calculation: The derivative of the given function gives the slope of the tangent line. Apply the product rule of differentiation for the given function to get the derivative function f ′ ( x ) . f ′ ( x ) = d d x [ ( x 2 + 1 ) ( 2 − x ) ] = ( x 2 + 1 ) ⋅ d d x ( 2 − x ) + ( 2 − x ) ⋅ d d x ( x 2 + 1 ) = ( x 2 + 1 ) ⋅ ( − 1 ) + ( 2 − x ) ⋅ ( 2 x ) = − x 2 − 1 + 4 x − 2 x 2 ( Apply distributive property ) Further solve the above equation for f ′ ( x ) . f ′ ( x ) = − 3 x 2 + 4 x − 1 ( Combine like terms ) (1) The slope of tangent line is, m = f ′ ( x ) (2) The tangent is horizontal on the graph of function f ( x ) when the slope of line is zero that means m = 0

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 3.2, Problem 47E

To determine

To find: The points on the graph of given function f(x) at which the tangent line is horizontal.

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Chapter 3 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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The points on the graph of given function f ( x ) at which the tangent line is horizontal.

Solution Summary: The author explains how to find the tangent line at the points on the graph of given function f(x).

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To find: The points on the graph of given function f(x) at which the tangent line is horizontal.

Chapter 3 Solutions