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Local extreme points and inflection points Suppose that f
Chapter 7, Problem 86AE(choose chapter or problem)
Local extreme points and inflection points Suppose that f has two continuous derivatives at a.
a. Show that if f has a local maximum at a, then the Taylor polynomial \(p_{2}\) centered at a also has a local maximum at a.
b. Show that if f has a local minimum at a, then the Taylor polynomial \(p_{2}\) centered at a also has a local minimum at a.
c. Is it true that if f has an inflection point at a, then the Taylor polynomial \(p_{2}\) centered at a also has an inflection point at a?
d. Are the converses to parts (a) and (b) true? If \(p_{2}\) has a local extreme point at a, does f have the same type of point at a?
Questions & Answers
QUESTION:
Local extreme points and inflection points Suppose that f has two continuous derivatives at a.
a. Show that if f has a local maximum at a, then the Taylor polynomial \(p_{2}\) centered at a also has a local maximum at a.
b. Show that if f has a local minimum at a, then the Taylor polynomial \(p_{2}\) centered at a also has a local minimum at a.
c. Is it true that if f has an inflection point at a, then the Taylor polynomial \(p_{2}\) centered at a also has an inflection point at a?
d. Are the converses to parts (a) and (b) true? If \(p_{2}\) has a local extreme point at a, does f have the same type of point at a?
ANSWER:Solution 86AE
Step 1:
a. Show that if f has a local maximum at a, then the Taylor polynomial p2centered at a also has a local maximum at a.
We know that the taylor polynomial of order 2 centered at a is given by,
Given that has local maximum at
Therefore by definition of local maximum, we have
To prove : has local maximum at a
We have
(since )
Thus
Since
Thus
Therefore and
By definition of local maximum, we say that has local maximum at a
Hence if f has a local maximum at a, then the Taylor polynomial centered at a also has a local maximum at a.