Rolle's theorem with limits at infinity

Let the function $f(x)$ have a finite derivative $f'(x)$ at every point of the finite or infinite interval $(a,b)$, and let

Prove that

where $c$ is some point in the interval $(a,b)$.

Prove that if the function $f(x)$ is twice differentiable on the infinite interval $(x_{0},+\infty )$, and

then there is at least one point $\xi$ in the interval $(x_{0},+\infty )$ satisfying

Prove that if the function $f$ is differentiable on a finite or infinite interval $(a ; b)$ and there exist equal finite limits, or infinite limits of the same sign, $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$, then there exists a point $\xi \in (a ; b)$ such that $f^{\prime }(\xi ) = 0$.

Let the function $f$ be differentiable for all $x \in \mathbb {R}$ and let $\lim_{x \rightarrow \infty } f(x) = 0$. Prove that there exists a point $\xi$ such that $f^{\prime }(\xi ) = 0$.

Let the function $f(x)$ have a continuous second-order derivative on $\mathbb {R}$, and suppose finite limits of this function exist as $x \rightarrow -\infty$ and $x \rightarrow +\infty$. Prove that there is a point $x_{0}$ such that $f^{\prime }\left(x_{0}\right) = 0$.

Let $f \in C^{\infty }([0, \infty ))$, $f(0)=\lim_{x \rightarrow +\infty } f(x)$. Prove that for each $n \in \mathbb {N}$ the function $f^{(n)}$ vanishes.