Cubic Spline: Prove S(3/2) = (3/2)^3 Let S(x) be the not-a-knot cubic spline interpolant of the points (0, 0), (1, 1), (2, 8), and (3, 27). Explain why $S(3/2) = (3/2)^3$ .

The not-a-knot condition requires the third derivative of the interpolant is continuous at point (1, 1) and at point (2, 8). This means the 3 cubic polynomial segments actually come from the same cubic polynomial. Therefore, you can conclude that $S(x)=x^3$ and therefore $S(3/2)=(3/2)^3$.