There are two curves $p_1$ and $p_2$ given by $y= ax^2+bx+c$ and $y=dx^2+ex+f$ respectively. Also the curves $p_1$ and $p_2$ pass through $(x_1,y_1)$ and $(x_2,y_2)$ respectively.
Now I would like to draw a curve ($y=f(x)$) which connects the curves $p_1$ and $p_2$ with the following conditions:
- The connecting curve should pass through $(x_1,y_1)$ and $(x_2,y_2)$
- When the three curves are considered together, should have a second derivative at $(x_1,y_1)$ and $(x_2,y_2)$
- The connecting curve should be $C^2$ function (differentiable twice)
What I have done so far: From second condition, I have $g''(x1) = 2a$ and $g''(x2)=2d$.
I see that these two are different constants. So, Is it even possible to draw a curve with such a conditions? If so, how do I move from here?