Matrice si functii derivabile
Posted: Wed Apr 08, 2009 6:45 pm
Fie \( {\rm f:[a,b]} \to {\rm R} \) o functie de n ori derivabila si \( a = x_0 < x_1 < ... < x_n = b \). Aratati ca \( \exists {\rm c} \in {\rm (a,b)} \) astfel incat
\(
\left( {\begin{array}{ccc}
1 & 1 & {...} & 1 \\
{x_0 } & {x_1 } & {...} & {x_n } \\
{...} & {...} & {...} & {...} \\
{f(x_0 )} & {f(x_1 )} & {...} & {f(x_n )} \\
\end{array}} \right) = \frac{1}{{n!}}f^{(n)} (c)\prod\limits_{1 \le i < j \le n} {(x_j - x_i )}. \)
\(
\left( {\begin{array}{ccc}
1 & 1 & {...} & 1 \\
{x_0 } & {x_1 } & {...} & {x_n } \\
{...} & {...} & {...} & {...} \\
{f(x_0 )} & {f(x_1 )} & {...} & {f(x_n )} \\
\end{array}} \right) = \frac{1}{{n!}}f^{(n)} (c)\prod\limits_{1 \le i < j \le n} {(x_j - x_i )}. \)