a. Explanation Proof: Suppose that f ″ ( x ) > 0 on the interval containing a . For every x in the interval the second order Taylor polynomial is f ( x ) = f ( a ) + f ′ ( c ) 1 ! ( x − a ) + f ″ ( c ) 2 ! ( x − a ) 2 . At the point a , the expansion is f ( x ) = f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 b. To determine To prove: If the second derivative f ″ ( x ) < 0 on some interval containing a , then f ( x ) has a local maximum at a .

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 11.3, Problem 91E

To determine

To prove: If the second derivative f″(x)>0 on some interval containing a, then f(x) has a local minimum at a.

To determine

To prove: If the second derivative f″(x)<0 on some interval containing a, then f(x) has a local maximum at a.

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Chapter 11.3, Problem 90E

Chapter 11.4, Problem 1QC

Chapter 11 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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To prove: If the second derivative f ″ ( x ) > 0 on some interval containing a , then f ( x ) has a local minimum at a .

Solution Summary: The author explains that if the second derivative f′′(x)>0 on some interval containing a, the center is the local minimum for the function.

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To prove: If the second derivative f″(x)>0 on some interval containing a, then f(x) has a local minimum at a.

To prove: If the second derivative f″(x)<0 on some interval containing a, then f(x) has a local maximum at a.

Want to see the full answer?

Chapter 11 Solutions

To prove: If the second derivative f ″ ( x ) &gt; 0 on some interval containing a , then f ( x ) has a local minimum at a .

Solution Summary: The author explains that if the second derivative f′′(x)>0 on some interval containing a, the center is the local minimum for the function.

Concept explainers

Videos

To prove: If the second derivative f″(x)>0 on some interval containing a, then f(x) has a local minimum at a.

To prove: If the second derivative f″(x)<0 on some interval containing a, then f(x) has a local maximum at a.

Want to see the full answer?

Chapter 11 Solutions

To prove: If the second derivative f ″ ( x ) > 0 on some interval containing a , then f ( x ) has a local minimum at a .