Taylor Formula and 2nd derivative test explained

Its Tuesday Night and I am updating this blog from Morris Library. I came to school today early morning and haven't gone back. I had lunch at the local Baptist Church, which sponsors free lunch for students. Its a good place to connect with friends and enjoy a good meal. I spent some time today going over the Taylor's formula. Taylor's formula is made up of two parts. The first part is P(n) and the second part is R(n). P(n) stands for polynomial and R(n) stands for Remainder. The n stands for number of derivatives. If the function f(x) on closed interval [a,b] is n times differentiable and has (n+1)th derivative existing and continious on open interval (a,b) then it can be written as a sum of P(n) and R(n). If it has all derivatives then it can be written as sum(f(c)^(n)*(x-c)^n,n = 0..infinity). Once you realized this its easy to see how 2nd derivative test work. For 2nd derivative test it should have both first and second derivative to exist and continuous over the interval [a,b]. At critical pt c. f(c)^(1) is zero. However the remainder (x-c)^2 is always positive. So the sign depends upon f(c)^(2) and depending if its positive or negative we can tell if the function has minima or maxima