# Find polynomial with specific constraints [closed]

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:

1. The polynomial degree must be $$2 \leq d \leq 8$$.
2. For every $$a_i$$ there holds $$L_{min} <= a_i \leq= L_{max}$$
3. For every $$L_{min} \leq x \leq L_{max}$$ every $$L_{min} \leq f(x) \leq L_{max}$$
• Maybe constraint (3) should read: $L_{min} \leq f(x) \leq L_{max}$ For every $x$ such that $L_{min} \leq x \leq L_{max}$ Nov 4, 2018 at 3:44
• It looks like any increasing polynomial with non-zero positive coefficients would satisfy your 3rd requirement, given that the first 2 are satisfied. Nov 4, 2018 at 3:50

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).