How to parameterize the maximum of a function? $f(x) := -4(\frac{1}{2}-x)^2+1$
Is it possible to construct a parameterized version of $f$, say $f_a$, which fulfills:


*

*$f_a(0) = f_a(1) = 0$

*$f_a(a) = 1$

*$f_a'(x) = 0 \Leftrightarrow x=a$

*$f_a''(a) < 0$

*$f_{1/2} = f$

*strictly monotonically increasing on [0,a)

*strictly monotonically decreasing on (a,1]

*$f_a$ does not have to be a polynomial necessarily

*$f_a$ is differentiable on [0,1]



I think you get the idea - just imagine a function like $f$ and visually shift the maximum along the red line while keeping anything else as (subjectively) constant as possible.
I would also be interested in learning whether this can be done numerically or if there is a systematic analytical method to achieve something like this. So I could apply this method to more complex functions. Also a definite "this is not possible because ..." would be a welcome answer.

$$f(x)=\begin{cases}1-(x-a)^2/a^2, x<a\\1-(x-a)^2/(1-a)^2, x\ge a\end{cases}$$
It is differentiable, but unfortunately it has no second derivative at $x=a$.
