22. Show that if $Q \in \mathbb { R } [ X ]$ is a polynomial such that $\operatorname { deg } ( Q ) < \operatorname { deg } ( P )$ and, for every integer $1 \leqslant i \leqslant m$ and every integer $0 \leqslant k < k _ { i } , Q ^ { ( k ) } \left( t _ { i } \right) = 0$, then $Q = 0$.\\
2b. Show that there exists a unique polynomial $H ( f , P ) \in \mathbb { R } [ X ]$ such that $\operatorname { deg } ( H ( f , P ) ) < \operatorname { deg } ( P )$ and such that, for every integer $1 \leqslant i \leqslant m$ and every integer $0 \leqslant k < k _ { i }$,

$$H ( f , P ) ^ { ( k ) } \left( t _ { i } \right) = f ^ { ( k ) } \left( t _ { i } \right) .$$

For $t \in [ a , b ] \backslash \left\{ t _ { 1 } , \ldots , t _ { m } \right\}$. We set

$$Q ( f , P ) ( t ) = \frac { f ( t ) - H ( f , P ) ( t ) } { \left( t - t _ { 1 } \right) ^ { k _ { 1 } } \cdots \left( t - t _ { m } \right) ^ { k _ { m } } } .$$

3a. We set $g = f - H ( f , P )$. Show that, for every integer $1 \leqslant i \leqslant m$ and every real $x \in [ a , b ]$, we have

$$f ( x ) - H ( f , P ) ( x ) = \left( x - t _ { i } \right) ^ { k _ { i } } \int _ { 0 } ^ { 1 } \frac { v ^ { k _ { i } - 1 } } { \left( k _ { i } - 1 \right) ! } g ^ { \left( k _ { i } \right) } \left( t _ { i } v + x ( 1 - v ) \right) d v$$

3b. Show that the function $Q ( f , P )$ extends uniquely to a function of class $\mathscr { C } ^ { \infty }$ from $[ a , b ]$ to $\mathbb { R }$.

4a. Let $s _ { 0 } \in [ a , b ]$ and let an integer $n \geqslant 1$. Show that

$$Q \left( f , \left( X - s _ { 0 } \right) ^ { n } \right) \left( s _ { 0 } \right) = \frac { f ^ { ( n ) } \left( s _ { 0 } \right) } { n ! }$$

\begin{itemize}
  \item 4b. Let $P _ { 1 } , P _ { 2 } \in \mathbb { R } [ X ]$ be two monic polynomials split in $] a , b [$. Show that
\end{itemize}

$$H \left( f , P _ { 1 } P _ { 2 } \right) = H \left( f , P _ { 1 } \right) + P _ { 1 } H \left( Q \left( f , P _ { 1 } \right) , P _ { 2 } \right) \quad \text { and } \quad Q \left( f , P _ { 1 } P _ { 2 } \right) = Q \left( Q \left( f , P _ { 1 } \right) , P _ { 2 } \right)$$

We fix $t \in [ a , b ] \backslash \left\{ t _ { 1 } , \ldots , t _ { m } \right\}$. For all $s \in [ a , b ]$, we set

$$Q _ { t } ( s ) = f ( s ) - H ( f , P ) ( s ) - Q ( f , P ) ( t ) \prod _ { i = 1 } ^ { m } \left( s - t _ { i } \right) ^ { k _ { i } }$$

5a. Show that the function $Q _ { t }$ vanishes to order $\operatorname { deg } ( P ) + 1$ in the interval [ $\min \left( t , t _ { 1 } \right) , \max \left( t , t _ { m } \right)$ ].\\
5b. Deduce that if $P$ is monic, there exists $\xi \in \left[ \min \left( t , t _ { 1 } \right) , \max \left( t , t _ { m } \right) \right]$ such that

$$f ( t ) - H ( f , P ) ( t ) = \frac { f ^ { ( \operatorname { deg } ( P ) ) } ( \xi ) } { \operatorname { deg } ( P ) ! } P ( t )$$

We say that a function $h$ from $[ a , b ]$ to $\mathbb { R }$ is absolutely monotone on an interval $[ a , b ]$ if it is of class $\mathscr { C } ^ { \infty }$ on $[ a , b ]$ and if, for every integer $n \geqslant 0$, the function $h ^ { ( n ) }$ takes positive values on $[ a , b ]$. In particular $h$ takes positive values.\\