Explanation Given information: The function is f ( x ) = 1 + x + x 2 x 2 + x − 2 . Formula Used: Let the function f and f ′ are differentiable on an interval L. If f ″ is positive, then f is concave up. If f ″ is negative, then f is concave down. Let the function f and f ′ are differentiable. Also, Let c be a critical point of a function f with f ′ ( c ) = 0 . Then, f ″ is continuous on an interval. If f ″ ( c ) is positive, then f has a local minimum at x = c . If f ″ ( c ) is negative, then f has a local maximum at x = c . If f ″ ( c ) is 0, then test fails. Calculation: The function is f ( x ) = 1 + x + x 2 x 2 + x − 2 . Find the derivative of the function. f ′ ( x ) = d d x ( 1 + x + x 2 x 2 + x − 2 ) = ( x 2 + x − 2 ) d d x ( 1 + x + x 2 ) − ( 1 + x + x 2 ) d d x ( x 2 + x − 2 ) ( x 2 + x − 2 ) 2 = ( x 2 + x − 2 ) ( 1 + 2 x ) − ( 1 + x + x 2 ) ( 2 x + 1 ) ( x 2 + x − 2 ) 2 = − 3 ( 2 x + 1 ) ( x 2 + x − 2 ) 2 Substitute − 3 ( 2 x + 1 ) ( x 2 + x − 2 ) 2 = 0 . − 3 ( 2 x + 1 ) ( x 2 + x − 2 ) 2 = 0 − 3 ( 2 x + 1 ) = 0 x = − 1 2 The derivative f ′ is 0 at point − 1 2

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 3.3, Problem 31S

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Find the local extrema of function f by the second-derivative test. If it fails, then use the first-derivative test. Also, draw the graph.

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Chapter 3.3, Problem 30S

Chapter 3.3, Problem 32S

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Find the local extrema of function f by the second-derivative test. If it fails, then use the first-derivative test. Also, draw the graph.

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Find the local extrema of function f by the second-derivative test. If it fails, then use the first-derivative test. Also, draw the graph.

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