
Finding a parabola
Consider the function f(x) which is defined to be x when x < 0 and 2x when x> 0. This function has a corner at the orgin and so is not smooth there. We could replace f by a smooth function g if we could find a parabola that is tangent to f at two points. The new function g would be defined to agree with f, except between the two points of tangency where it would agree with the parabola.
More than one parabola would do for this purpose. Big ones would give smoother transitions than small ones. Find any one.
As always any help/hints are greatly appreciated!

We know, first of all, that such a parabola would be defined by a function of the form
$\displaystyle h(x)=ax^2+bx+c.$
Now, all we need to do is pick two points: say, $\displaystyle x=1$ and $\displaystyle x=1$. From the statement of the problem, we need
$\displaystyle h(1)=1\;\;\;\;\;\;\;\;\;\;h(1)=2\;\;\;\;\;\;\;\;\;\; h'(1)=1\;\;\;\;\;\;\;\;\;\;h'(1)=2$
This gives us enough information to solve an algebraic equation for $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$.