Say I am given a degree $d$, an upper and a lower limit $L_{min}$ and $L_{max}$ and a constant $k$ with $L_ {min} \leq k \leq L_ {max}$. How can I find a polynomial in the form of $f(x)=k + a_1x^1+a_2x^2+a_3x^3 \ldots + a_dx^d$ for which the following properties hold:
- The polynomial degree must be $2 \leq d \leq 8$.
- For every $a_i$ there holds $L_{min} <= a_i \leq= L_{max}$
- For every $L_{min} \leq x \leq L_{max}$ every $L_{min} \leq f(x) \leq L_{max}$
Such a polynomial might not exist. For example, suppose $L_{min}=0$ and $L_{max}=1$ and $k=0$. Then for every polynomial $f(x)=k+a_1x+\ldots +a_dx^d$, one has $f(1)=k+a_1+\ldots+a_d \geq k+a_d \geq 2$, which contradicts condition (3).