# Find a cubic hermite interpolation of the function

Let be $I=[0,2]\subset\mathbb{R}$ and $f:I\to\mathbb{R}$ with

$$f(x):=\frac{x^2-5}{-x^3+x^2-4}.$$

Define a polynomial $p$ using the cubic Hermite interpolation method with the grid points $x_0=0,x_1=1$ and $x_2=2$. The question is: how many times is $f$ continuously differentiable on $I$?

My idea is: I calculate

$$f(x_0)=\frac{5}{4}, f(x_1)=1, f(x_2)=\frac{1}{8}.$$

How can I define $p$?