Explanation Given information: The function h ( x ) is as shown below, h ( x ) = x ( x 2 − 1 ) x 2 + 1 Concept used: If a function f ( x ) has a critical point as x = c then, f ( c ) is a local maximum if f ′ ( x ) changes sign from positive to negative at x = c . f ( c ) is a local minimum if f ′ ( x ) changes sign from negative to positive at x = c . If a function f ( x ) is continuous and differentiable in the interval ( a , b ) then, f ( x ) is an increasing function on ( a , b ) if f ′ ( x ) > 0 for all x ∈ ( a , b ) . f ( x ) is a decreasing function on ( a , b ) if f ′ ( x ) < 0 for all x ∈ ( a , b ) . Calculation: Consider the function as f ( x ) . The derivative of the function f ( x ) is given as, h ( x ) = x ( x 2 − 1 ) x 2 + 1 f ′ ( x ) = x ( x 2 − 1 ) x 2 + 1 Draw the graph of the function h ( x ) = x ( x 2 − 1 ) x 2 + 1 . Solve the equation h ( x ) = 0 to obtain the roots of the function as shown below, x ( x 2 − 1 ) x 2 + 1 = 0 x ( x 2 − 1 ) = 0 The solution of the above equation is x = 0 , x = 1 and x = − 1 . Therefore, the roots of the function h ( x ) = x ( x 2 − 1 ) x 2 + 1 are x = 0 , x = 1 and x = − 1 . Figure 1 represents the graph of the function h ( x ) = x ( x 2 − 1 ) x 2 + 1 . The critical point is defined as the point or the value of x such that f ′ ( x ) = 0 . This implies that the critical points for the function f ( x ) are x = − 1 , x = 0 and x = 1 . The critical points divide the domain of the function into four intervals as ( − ∞ , − 1 ) , ( − 1 , 0 ) , ( 0 , 1 ) and ( 1 , ∞ ) . The behavior in value of the function f ′ ( x ) is expressed in Table 1 as shown below, Interval Test value Sign of f ′ ( x ) ( − ∞ , − 1 ) f ′ ( − 2 ) = − 6 5 Negative ( − 1 , 0 ) f ′ ( − 0.5 ) = 3 10 Positive ( 0 , 1 ) f ′ ( 0

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 4.3, Problem 54E

To determine

To sketch: The graph of the function h(x)=x(x2−1)x2+1 and f(x) such that f′(x)=h(x) and find the nature of the critical points for the function f(x).

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Chapter 4.3, Problem 53E

Chapter 4.3, Problem 55E

Chapter 4 Solutions

Calculus - Standalone book

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 graph of the function h ( x ) = x ( x 2 − 1 ) x 2 + 1 and f ( x ) such that f ′ ( x ) = h ( x ) and find the nature of the critical points for the function f ( x ) .

Solution Summary: The author sketched the graph of the function h(x) and the critical points for it.

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To sketch: The graph of the function h(x)=x(x2−1)x2+1 and f(x) such that f′(x)=h(x) and find the nature of the critical points for the function f(x).

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