Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $W = \prod_{i=1}^n (X - a_i)$ and let $f$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I$. Let $P = \Pi ( f )$ be the interpolation polynomial of $f$ associated with the real numbers $a _ { 1 } , \ldots , a _ { n }$, defined by $$\Pi ( f ) = \sum _ { i = 1 } ^ { n } f \left( a _ { i } \right) L _ { i }.$$ For all $x \in I$, show that there exists $c \in I$ such that $$f ( x ) - P ( x ) = \frac { f ^ { ( n ) } ( c ) } { n ! } W ( x ).$$ For $x$ distinct from the $a _ { i }$, one may consider the function $r$ defined on $I$ by $$r ( t ) = f ( t ) - P ( t ) - K W ( t )$$ where the real number $K$ is chosen so that $r ( x ) = 0$.
Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $W = \prod_{i=1}^n (X - a_i)$ and let $f$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I$. Let $P = \Pi ( f )$ be the interpolation polynomial of $f$ associated with the real numbers $a _ { 1 } , \ldots , a _ { n }$, defined by
$$\Pi ( f ) = \sum _ { i = 1 } ^ { n } f \left( a _ { i } \right) L _ { i }.$$
For all $x \in I$, show that there exists $c \in I$ such that
$$f ( x ) - P ( x ) = \frac { f ^ { ( n ) } ( c ) } { n ! } W ( x ).$$
For $x$ distinct from the $a _ { i }$, one may consider the function $r$ defined on $I$ by
$$r ( t ) = f ( t ) - P ( t ) - K W ( t )$$
where the real number $K$ is chosen so that $r ( x ) = 0$.