Chapter 3.4, Problem 4E

Find $k_1,k_2,k_3$ in the following cubic spline. Which of the three end conditions—natural, parabolically terminated, or not-a-knot—if any, are satisfied?

$S\left(x\right)=\{\begin{array}{rr}\hfill 4+k_{1}x+2x^2-\frac{1}{6}{x}^3& \hfill \text{on}\left[0,1\right]\\ \hfill \begin{array}{l}1-\frac{4}{3}\left(x-1\right)+k_2{\left(x-1\right)}^2-\frac{1}{6}{\left(x-1\right)}^3\\ 1+k_3\left(x-2\right)+{\left(x-1\right)}^2-\frac{1}{6}{\left(x-2\right)}^3\end{array}& \hfill \begin{array}{l}\text{on}\left[1,2\right]\\ \text{on}\left[2,3\right]\end{array}\end{array}$