How to determine if an equation represents a cubic spline? Given the equation 
$$
   f(x) = \left\{
     \begin{array}{lr}
       2x^3+x^2+4x+5 & : 0 \le x \le 1\\
       (x-1)^3 + 7(x-1)^2 + 12(x-1)+12 & : 1 \le x \le 2
     \end{array}
   \right.
$$
What is the process used to determine if this represents a cubic spline?
It's obviously a piecewise interpolation, and the second function simplifies, so perhaps it's better to write like:
$$
   f(x) = \left\{
     \begin{array}{lr}
       2x^3+x^2+4x+5 & : 0 \le x \le 1\\
       x^3+4x^2+x+6 & : 1 \le x \le 2
     \end{array}
   \right.
$$

Both expression are obviously cubic.  If they both have the same value and the same first and second derivatives at $x=1$, your set of expressions fits the definition of a cubic spline.  A pretty simple spline, but there it is.
