Let f be defined on [a, b], and let the nodes a = x0 < x1 <

Question:

Let f be defined on [a, b], and let the nodes a = x0 < x1 < x2 = b be given. A quadratic spline interpolating function S consists of the quadratic polynomial
S0(x) = a0 + b0(x − x0) + c0(x − x0)2 on [x0, x1]
And the quadratic polynomial
S1(x) = a1 + b1(x − x1) + c1(x − x1)2 on [x1, x2],
Such that
i. S(x0) = f (x0), S(x1) = f (x1), and S(x2) = f (x2),
ii. S ∈ c1[x0, x2].