How to prove with Taylor-theorem that if $f$ and $f''$ are bounded, than $f'$ is bounded, too?

Question

$f:R→R$ is twice differentiable. $f$ and $f''$ are bounded. How to prove with Taylor-theorem that $f'$ is bounded as well?

Answer 1

For any $x$, by Taylors theorem, $$f(x+1) = f(x) + f^\prime (x) + \frac{1}{2}f^{\prime\prime}(\xi)$$

for some $\xi\in (x, x+1)$. Now solve for $f^\prime(x)$.