Correct Answer - Option 1 : Constant
If f(x) is a polynomial of degree n then the nth divided difference of f(x) is a constant.
Proof:
The first divided difference of f(x) = xn
\(f(x_0,x_1)=\frac{(x_0+h)^n-x_0^n}{x_0+h-x_0}\)
\(f(x_0,x_1)=\frac{x_0^n+nh~x_0^{n-1}+.....h^n-x_0^n}{h}\)
\(f(x_0,x_1)=\frac{nh~x_0^{n-1}+.....h^n}{h}\)
The second divided difference can be given as:
\(f(x_0,x_1,x_2)=\frac{f(x_1,x_2)-f(x_0,x_1)}{x_2-x_0}\)
The kth divided difference can be given as:
\(f(x_0,x_1,x_2,...x_k)=\frac{f(x_1,x_2,...x_k)-f(x_0,x_1,...x_{k-1})}{x_k-x_0}\)
The divided difference operator is a linear operator.
The first divided difference is an (n-1)th degree polynomial.
∴ The second divided difference is an (n-2)th degree polynomial and hence the nth divided difference is (n-n)th degree polynomial that is a constant.
