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 .

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

2nd Edition

ISBN: 9780321954237

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

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 9.3, Problem 89E

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 88E

Chapter 9.3, Problem 90E

Students have asked these similar questions

53,85÷1,5=

3. In the space below, describe in what ways the function f(x) = -2√x - 3 has been transformed from the basic function √x. The graph f(x) on the coordinate plane at right. (4 points) -4 -&- -3 -- -2 4 3- 2 1- 1 0 1 2 -N -1- -2- -3- -4- 3 ++ 4

2. Suppose the graph below left is the function f(x). In the space below, describe what transformations are occuring in the transformed function 3ƒ(-2x) + 1. The graph it on the coordinate plane below right. (4 points)

Chapter 9 Solutions

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

Ch. 9.1 - Prob. 1QC Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - Prob. 5QC Ch. 9.1 - In Example 7, find an approximate upper bound for...Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E

Additional Math Textbook Solutions

Find more solutions based on key concepts

4. Interpreting P-value The Ericsson method is one of several methods claimed to increase the likelihood of a b...

Elementary Statistics (13th Edition)

Applying the Empirical Rule with z-Scores The Empirical Rule applies rough approximations to probabilities for ...

Introductory Statistics

Using the Empirical Rule In Exercises 29–34, use the Empirical Rule. 34. The monthly utility bills for eight ho...

Elementary Statistics: Picturing the World (7th Edition)

Fill in each blanks so that the resulting statement is true. Any set of ordered pairs is called a/an _______. T...

College Algebra (7th Edition)

Plot the point whose rectangular coordinates are ( 3,1 ) . What quadrant does the point lie in?

Precalculus

CHECK POINT I You deposit $1000 in a saving account at a bank that has a rate of 4%. a. Find the amount, A, of ...

Thinking Mathematically (6th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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.

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 9 Solutions

Additional Math Textbook 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 9 Solutions

Additional Math Textbook Solutions

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