The following questions pertain to the use of piecewise polynomials for interpolating a function f over nodes {i} 0- (a) Consider the n = 2 case: given nodes {xo, 1, 2}, define the piecewise polynomial P(x) by P(x) = Spo(x), x= [To, x1], [P₁(x), x = [₁, ₂], where po, Pi are polynomials in their respective intervals. Suppose we require the following: P match f at each of the nodes, P' match f' at each of the nodes, • P be continuous P' be continuous Can we find such a piecewise polynomial, and if so what order polynomials should po, p₁ be? Hint: Does the number of conditions equal the total number coefficients for P? (b) Generalize your work from part (a) given nodes {x;}=0.

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