Need just a little bit of help proving this graph is not a natural cubic spline

So this is my graph

I am aware that we can see its not a cubic spline because $$p_1(x), 0≤x≤1$$ is linear. But I wanted to prove it and got stuck.

From what I know, For it to be a natural cubic spline, the second derivatives of the endpoints needs to be 0. Endpoints are at x=0 and x=3. I set up my equations accordingly.

This is were I get stuck. I dont know what I should do next.