Taylor and Maclaurin Polynomials in Function Analysis

If a function is even, its odd-order derivatives are odd, and its even-order derivatives are even.
Similarly, if a function is odd, its odd-order derivatives are even, and its even-order derivatives are odd.

Given an n-th order Taylor Expansion of an n-times differentiable function, if the function at that point is equal to zero, and all its derivatives are zero up until a derivative of order , then

Suppose a function f(x) is n-times differentiable at a point a.
If you have zeroes for all parts of the expansion until a non-zero m-th order derivative (so, if everything is zero until some derivative of order m), for m at least 2:

• If this m is even, this point a is a local extremum point. Maximum if , and minimum if

• If this m is odd, the point a is an inflection point. Descending inflection point if , and ascending if

You survived.
Calculus Interlude - A Final Review