If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

Question

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

Hi, I need help with the above problem. I'm working through a previous exam for practice for my final exam. Unfortuntately, I don't have any answer solutions, so I would really appreciate it if someone can help me with this problem.

I do not know how to solve it. But, it makes sense to me intuitively because $f$ being analytic with its derivatives equal to $0$ at a point implies $0$ slope, and $0$ slope of the slope (and so on), meaning that $f$ cannot really move from there, which means it must be identically $0$. I don't know how to prove this, though. I don't know why it's necessary for $f$ to be analytic for this to be true either.

Answer 1

Analytic means you have a Taylor series. Write down the Taylor series.