Is there a general information on a polynomial with minimums in two given points

Question

I am thinking about a description of an object with a function $f(x)$ such that it has minimums at $x=0$ and $x=1$. This function will be represented by a polynomial. It is not difficult to write down such a polynomial in specific cases, for example, the polynomial of the smallest degree has the following form:

$$\frac1{12} x^2 (a (6 - 4 x) + x (-4 + 3 x))$$

with $0<a<1$. However, there can be plenty of such polynomials.

Please help me to find out if any general facts are known about such polynomials.

Answer 1

It is known that if a function has a continuous second derivative, then the condition "$f$ has a strong local minimum in $x$" is equivalent to "$(f^{'}(x) = 0)\&(f^{''}(x) > 0)$" (this fact is a direct consequence of Taylor theorem). Polynomials do have a continuous second derivative as $(\sum_{i = 0}^n a_ix^i)^{'}= \sum_{i = 1}^{n} ia_ix^{i - 1}$. So the condition $\sum_{i = 0}^n a_ix^i$ has strong local minima in $0$ and $1$ is equivalent to the following inequalities on its coefficients:

$$\begin{cases} a_1 = 0 \\ 2a_2 > 0 \\ \sum_{i = 1}^n ia_i = 0 \\ \sum_{i = 2}^n i(i-1)a_i > 0 \end{cases}$$

which can be simplified to:

$$\begin{cases} a_1 = 0 \\ a_2 > 0 \\ \sum_{i = 2}^n ia_i = 0 \\ \sum_{i = 2}^n i(i-1)a_i > 0 \end{cases}$$

From this system of two equalities and two inequalities you can derive everything you need about those polynomials.

For example, this system can be used for classification of the minimal degree polynomials with that property:

That is the case when $n = 4$. As this property is preserved by addition of constant and multiplication by constant we can without the loss of generality assume that $a_0 = 0$ and $a_4 \in \{-1; 1\}$.

If $a_4 = 1$, then

$$\begin{cases} a_2 > 0 \\ 2a_2 + 3a_3 + 4 = 0 \\ 2a_2 + 6a_3 + 12 > 0 \end{cases}$$

$$\begin{cases} a_2 > 0 \\ a_3 = \frac{-2(2 + a_2)}{3} \\ -2a_2 + 4 > 0 \end{cases}$$

From that it follows $$\begin{cases} a_2 \in (0; 2) \\ a_3 = \frac{-2(2 + a_2)}{3} \\ -2a_2 + 4 > 0 \end{cases}$$

If $a_4 = -1$ then using the same method you can conclude, that there are no such polynomials with two strong local minima.

Thus such polynomials of minimal degree are exactly $ax^2(3x^2 - 2(2+b)x + 3b) + c$, where $a \in \mathbb{R}_+$, $b \in (0; 2)$, $c \in \mathbb{R}$.

Answer 2

HINT.- For your particular case you can do $$\begin{cases}f'(x)=x(x-1)p(x)\\f''(x)=(x-1)p(x)+xp(x)+x(x-1)p'(x)\end{cases}$$ where $p(x)$ is a polynomial and $$\begin{cases}f''(0)=-p(0)\gt0\\f''(1)=p(1)\gt0\end{cases}$$ This gives the general solution making an easy integration after chosing adequate polynomial $p(x)$.