How to show the following equality for twice differentiable function?

Question

Let $f$ be a $C''$ function on $(a, b)$ and suppose there is a point $c$ in (a, b) with $$f(c)= f'(c)=f''(c) = 0$$ Show that there is a continuous function $h$ on $(a, b)$ with $$f(x) =(x-c)^2h(x)$$ for all $x$ in $(a, b)$.

Answer 1

Define $h(x) = \frac{f(x)}{(x-c)^2}$ for all $x \in (a,b)$ different from $c$. Then try to show that $h$ can be extended by continuity at $x = c$ using the hypothesis.

Answer 2

hint

For $x$ in $(a,b)$,

$$f(x)=f(c)+(x-c)f'(c)+\frac{(x-c)^2}{2}f''(c)+(x-c)^2\epsilon(x)$$