Suppose f has continuous first and second derivatives at a.

a. Show that if f has a local maximum at a, then the Taylor polynomial p₂ 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 p₂ 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 p₂ centered at a also has an inflection point at a?

d. Are the converses in parts (a) and (b) true? If p₂ has a local extreme point at a, does f have the same type of point at a?