For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$ Deduce that for every $x > 0$, for every $n \in \mathbb{N}^*$, there exists a $y_n \in \left]0, n\right[$ such that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ One may proceed by induction on $n \in \mathbb{N}^*$ and use the three preceding questions.
For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application
$$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Deduce that for every $x > 0$, for every $n \in \mathbb{N}^*$, there exists a $y_n \in \left]0, n\right[$ such that
$$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$
One may proceed by induction on $n \in \mathbb{N}^*$ and use the three preceding questions.