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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 $L_{min} \leq f(x) \leq L_{max}$
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  • $\begingroup$ Maybe constraint (3) should read: $L_{min} \leq f(x) \leq L_{max}$ For every $x$ such that $L_{min} \leq x \leq L_{max}$ $\endgroup$
    – NoChance
    Nov 4, 2018 at 3:44
  • $\begingroup$ It looks like any increasing polynomial with non-zero positive coefficients would satisfy your 3rd requirement, given that the first 2 are satisfied. $\endgroup$
    – NoChance
    Nov 4, 2018 at 3:50

1 Answer 1

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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).

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  • $\begingroup$ I guess this is only on case where the polynomial does not exist, and it does not prove that that it can't exist in general. for example, a second degree equation that has a1, a2, a3 all non zero and >0, between 1 and 10 satisfies the constraints. $\endgroup$
    – NoChance
    Nov 4, 2018 at 3:06

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