Let's say I have this general 3rd order polynomial defined:
$f(x):=ax^3+bx^2+cx+d$
If I have concrete constraints of x and y values, I am able to solve the parameters. For example:
- $f(0)=0$
- $f(1)=0$
- $f'(0)=1$
- $f'(1)=-1$
gives the solution of $a = 0$, $b = -1$, $c = 1$ and $d = 0$.
Let's say, though, that I do not have such concrete points of the poynomial and its derivatives, and instead a general statement, such as $0 \leq f(x) < 5$ for $x \in [0, 1] $?
Is it still possible to calculate parameters or a set of parameters that matches these constraints?