How can we use interpolation to create two functions?

[haya]

Homework Statement

[PLAIN]http://im2.gulfup.com/2011-04-01/1301652393881.gif

Homework Equations

how can we do into 2 functions

The Attempt at a Solution

[PLAIN]http://im2.gulfup.com/2011-04-01/1301652475262.gif

 

[lurflurf]

I do not understand what you are trying to do with all those L's. Just let x=x0,x1,.,xn to verify the equation.

 

[haya]

how can i show that ?

 

[lurflurf]

let
g(x)=-f(x)+p(x)(x_n-x)/(x_n-x₀)+q(x)(x-x₀)/(x_n-x₀)
we want to prove that
g(x)=0 for x=x₀,x₁,.,x_n
given
p(x)-f(x)=0 for x=x₀,x₁,.,x_n-1
q(x)-f(x)=0 for x=x₁,x₂,.,x_n

hint
use the fact that 1=(x_n-x)/(x_n-x₀)+(x-x₀)/(x_n-x₀)

to write g(x) as
g(x)=-f(x)(x_n-x)/(x_n-x₀)-f(x)(x-x₀)/(x_n-x₀)
+p(x)(x_n-x)/(x_n-x₀)+q(x)(x-x₀)/(x_n-x₀)