Find $f(x)$ given $f(0), f(1)$ and $f[x1,x2,x3]$

Question

I need to find f(x) given

$f(0) = 0$, $f(1) = 2$, and the divided difference $f[x_1,x_2,x_3] = 1$ for any three points $x_1, x_2, x_3$

How do I go about solving this?

Answer 1

From the definition of divided differences, $f[x_0, x_1, x_2] =\frac{f(x_0)}{(x_0-x_1)(x_0-x_2)} +\frac{f(x_1)}{(x_1-x_0)(x_1-x_2)} +\frac{f(x_2)}{(x_2-x_0)(x_2-x_1)} $.

Putting $x_0 = 0$, $x_1 = 1$, $x_2 = x$, $f(0)=1$ and $f(1) = 2$,

$\begin{array}\\ 1 &=f[0, 1, x]\\ &=\frac{f(0)}{(0-1)(0-x)} +\frac{f(1)}{(1-0)(1-x)} +\frac{f(x)}{(x-0)(x-1)}\\ &=\frac{0}{x} +\frac{2}{(1-0)(1-x)} +\frac{f(x)}{x(x-1)}\\ &=\frac{2}{(1-x)} +\frac{f(x)}{x(x-1)}\\ \end{array} $

so $f(x) =x(x-1)+2x =x^2+x $.