Let S be a cubic spline that has knots t0 < t1 < · · · < tn. Let S be a cubic spline that has knots $t_0 < t_1 < · · · < t_n$. Suppose that
on the two intervals $[t_0, t_1]$ and $[t_2, t_3]$, S reduces to linear polynomials. What does
the polynomial S look like on the interval $[t_1, t_2]$ (linear, quadratic, or cubic)?
I feel like this may be asking me something trivial since (cubic spline) might imply something cubic i.e. $x^3$.. I'm not sure though. I know that the choice of degree most frequently made for a spline function is 3. I think I also read that by Taylor's Theorem, the two interval polynomials are forced to be the same, though I could of misread. Any advice or links would be great, this is sort of a vague topic I'm guessing

The shape of S within interval $[t_1,t_2]$ could be linear, quadratic or cubic. Each "segment" of this cubic spline in between two knots is a cubic polynomial by its own. Therefore, the shape could look linear, quadratic or cubic depending on the control points configuration. If your cubic spline is required to be $C^1$ continuous, then the shape of S within $[t_1,t_2]$ will not be linear unless its neighboring segments are collinear.
