Question: (Continuation of Problem) You may remember that the Second Derivative test tells us nothing when the second derivative is zero at the critical point. In this problem you will investigate that special case. Assume g has the same properties as in Problem 54, and that, in addition, g"(0) = 0. What does the Taylor polynomial tell you about whether g has a local maximum or minimum at x = 0?
Problem: All the derivatives of g exist at x = 0 and g has a critical point at x = 0.
(a) Write the nth Taylor polynomial for g at x = 0.
(b) What does the Second Derivative test for local maxima and minima say?
(c) Use the Taylor polynomial to explain why the Second Derivative test works.