found a function from equations and inequality?

Question

I'm Software engineer and I'm having little issue solving this problem let's called H.

Well I'm looking for the mathematical expression of the function f(x) based on 3 equations and one inequality. f(x) passes through 3 points and f(x) is a decreasing function when X€[0,END]

H = {f(0)=1, f(t)=α, f(50400)=0, f '(x)<0} and 0< t < END=50400, 0< a <1.

what i tried: I supposed the solution is polynomial of order 3 and can be expressed this way:

f(x)= ax(x-t)(x-END) + bx(x-t)+ cx+ d I found d,c,b still a !

1. Q1: Well am I doing this right ?
2. Q2: is there any theory I can use to find the solution ?

EDIT 1:

I know this have infinit solution so a solution with a lesser degree is preferable.

Answer 1

$$f(50400)=0$$ Leads to:

$$f(x)=g(x)(x-50400)$$

if you want or know that $f(x)$ is of degree $3$, then $g(x)$ is of degree $2$ and $f(x)$ is:

$$f(x)=(x^2+Bx+C)(x-50400)$$

Since $f(0)=1$, we have:

$$f(0)=(c)(-50400)=1 $$

Now $f(x)$ is:

$$f(x)= (x^2+Bx-\frac{1}{50400})(x-50400)$$

You say that $f(t)=a$, using this fact, you get the value of $B$ in terms of $a$.

Note: I don't know what you mean by "I found d,c,b still a" - Maybe you could apply the above steps on your function and get a.

f(x)= ax(x-t)(x-END) + bx(x-t)+ cx+ d I found d,c,b still a !

Answer 2

Quadratic is one option. There are infinitely many other options. Some examples: Scaled & shifted exponential function, $\frac{a}{x + b} - c$, etc.

Anyway, you can only have up to 3 degrees of freedom.