Taylor polynomial error of 3rd degree if f is only 2 times derivable.

Question

If f is only two times derivable and i know f's Taylor polynomial of second degree. Is the error even defined for that polynomial? Or it's equal to zero?

Answer 1

The error certainly exists, even if $f$ is only twice differentiable at $c$:

$\begin{align*} R(x) &= f(x) - \left( f(c) + f'(c) (x - c) + \frac{1}{2} f''(c) (x - c)^2 \right) \end{align*}$

What you might not be able to do is to use e.g. Lagrange's form of the remainder:

$\begin{align*} R(x) &= \frac{1}{3!} f'(\xi) (x - c)^3 \end{align*}$

for some $\xi$ between $c$ and $x$.

The derivation of this form of the remainder uses the mean value theorem for the derivative, and so assumes the relevant derivative is continuous in the relevant closed interval between $c$ and $x$.