Sequence and intermediate value theorem

Question

For part (a), By intermediate value theorem, there exist c between 0 and 1 such that $f(c) = 0$

Now, I supposed that there also exist d between 0 and 1 such that $ f(d) = 0 $ and $ c \neq d $ I am trying to get a contradiction but I am not sure how to use the fact that $f'(0) >0$ and $f''(x) \geq 0$

For part (b), I have tried using taylor's theorem and got

$f(x_n) = f(c) + f'(c)(x_n - c) + \frac{f''(\xi)(x_n - c)^2}{2}$ for some $\xi$ between $c$ and $x_n$

Substituting in what is given in the Question, f(c) disappears and I got

$x_n - x_{n+1} = \frac{f'(c)x_n - cf'(c)}{f'(x_n)} + \frac{(x_n-c)^2}{2 f'(x_n)} f''(\xi)$

Now I'm stuck

Thanks.