Let $f(x) = x^2\cdot (x-1)^2 \cdot (x-2)^2 \cdot (x-3)^2$. What is the piecewise cubic Hermite interpolant of $f$ on the grid $x_0 = 0$, $x_1 = 1$, $x_2 = 2$, $x_3 = 3$. Let $g(x) = ax^3 + bx^2 + cx +d$ for some parameters $a, b, c, d$ write down the piecewise cubic Hermite interpolation of g on the same grid.
I realize that $f$ and $f'$ are 0 at each node, so essentially the cubic polynomial that interpolates $f$ is just $g$. But I'm not sure of how to actually split $g$ into a piecewise cubic since it's already a cubic function.