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.