By "collaborate", I mean to give some instructions or constraints.

Consider the following curves: (as examples, only to illustrate my question in a better way):

Curve (I): Assume it is f instead of f':

My constraint are:

1. y[-4]==0, y[5.1]==0, y[10]==0
2. y'[x]>0 for x in (-∞,0) and (7,∞), and y'[x]<0 for x in (0,7)
3. y''[x]>0 for x in (5,∞), and y''[x]<0 for x in [-∞,5]

curve (II):

My constraints: such and such

Curve (III): Yes it is easy without mathematica, since these are just parts of the unit circle, but say we do not know.

Curve (IV): So many constraints, about y[x], about y'[x], and about y''[x]

I do not have great knowledge in Mathematica, but I believe it can help for such task.

If I have all constraints, such as intercepts, first and second derivatives, do I need to provide the image of these curves? Will providing support in finding a better fit? Will my constraints be sufficient?

Note that, the final result can be piecewise functions, as in Curve III

Can it handle many constraints? Curve IV really has many constraints, increasing and decreasing, concave up and concave down.

What if I need my constraints to be exact, with no deviation?

Your help would be appreciated. THANKS!

By "collaborate", I mean to give some instructions or constraints.

Consider the following curves: (as examples, only to illustrate my question in a better way):

Curve (I): Assume it is f instead of f':

My constraint are:

1. y[-4]==0, y[5.1]==0, y[10]==0
2. y'[x]>0 for x in (-∞,0) and (7,∞), and y'[x]<0 for x in (0,7)
3. y''[x]>0 for x in (5,∞), and y''[x]<0 for x in [-∞,5]

curve (II):

My constraints: such and such

Curve (III): Yes it is easy without mathematica, since these are just parts of the unit circle, but say we do not know.

Curve (IV): So many constraints, about y[x], about y'[x], and about y''[x]

I do not have great knowledge in Mathematica, but I believe it can help for such task.

If I have all constraints, such as intercepts, first and second derivatives, do I need to provide the image of these curves? Will providing support in finding a better fit? Will my constraints be sufficient?

Note that, the final result can be piecewise functions, as in Curve III

Can it handle many constraints? Curve IV really has many constraints, increasing and decreasing, concave up and concave down.

What if I need my constraints to be exact, with no deviation?

EDIT:

As suggested in the comment, I tried to clean and change the colour of Curve IV. Is this the way?

Secondly, would it be sufficient to provide only that? How to ensure local max and local min, ensure concavity, ensure intercepts, etc. (Note: I can provide such information, but with providing, will it be exact or just estimate?)

Your help would be appreciated. THANKS!