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want to find a cubic spline for f on the interval [a,b]

suppose we have n nodes with n-1 different intervals

I tried to find it using the Taylor expansion around any nodes say [tex]x_i \in [a,b][/tex]

[tex] f(x) - S(x) = f(x_i)-S(x_i) + (x-x_i)(f'(x_i)-S'(x_i))+ \frac{(x-x_i)^2(f''(x_i)-S''(x_i))}{2} + \frac{(x-x_i)^3(f'''(x_i)-S'''(x_i))}{3!} + \frac{(x-x_i)^4f^{(4)}(c)}{4!} [/tex]

i considered that the first three terms are zeros so

instead of x i sub [tex]x_{i+1} [/tex] since it is most sutiable x in the subinterval and we shall take the bound of the forth derivative of the function

ending with this

[tex]\mid f(x) - S(x) \mid = \frac{(x_{i+1}-x_i)^4 f^{(4)}(c)}{4!}[/tex]

is it right ?