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.

Step 2:

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.

We know that the taylor polynomial of order 2 centered at a is given by,

Given that has local minimum at

Therefore by definition of local minimum, we have

To prove : has local minimum at a

We have

(since )

Thus

Since

Thus

Therefore and

By definition of local minimum, we say that has local minimum at a

Hence if f has a local minimum at a, then the Taylor polynomial centered at a also has a local minimum at a.

Step 3:

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 ?

We know that the taylor polynomial of order 2 centered at a is given by,

Given that has inflection point at

Therefore by definition of inflection point, we have .

To prove : has inflection point at a

Set

Since f has inflection point at a, we get

change sign for negative and positive values of a.

Since we get

change sign for negative and positive values of a.

Therefore has inflection point at a.

Hence if f has a inflection point at a, then the Taylor polynomial centered at a also has a inflection point at a.