Problem 86AE

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 p2centered at a also has a local maximum at a.

b. Show that if f has a local minimum at a, then the Taylor polynomial p2centered 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 p2centered at a also has an inflection point at a ?

d. Are the converses to parts (a) and (b) true? If p2has a local extreme point at a, does f have the same type of point at a?

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.