For a quadratic function $f ( x )$, the function $g ( x ) = f ( x ) e ^ { - x }$ satisfies the following conditions. (가) The points $( 1 , g ( 1 ) )$ and $( 4 , g ( 4 ) )$ are inflection points of the curve $y = g ( x )$. (나) The number of tangent lines drawn from the point $( 0 , k )$ to the curve $y = g ( x )$ is 3 when $k$ is in the range $- 1 < k < 0$. Find the value of $g ( - 2 ) \times g ( 4 )$. [4 points]

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Let $\lambda$ be an eigenvalue of $U$ and $P$ an eigenvector associated with it.
VIII.F.1) Show that $P$ is a solution of a simple linear differential equation that we will specify.
VIII.F.2) What is the relationship between $\lambda$ and the degree of $P$?

Let $n$ in $\mathbb { N } ^ { * }$, verify that for real $x$
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( x ^ { n } \varphi _ { n } ( x ) \right) = x ^ { n } \varphi _ { n - 1 } ( x )$$

We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Using the power series development of $\varphi _ { n }$ (II.C.3), show that $\varphi _ { n }$ is a solution on $[ 0 , + \infty [$ of (III.1).

For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Show that the function $h$ is a solution of the differential equation $x \left( y ^ { \prime \prime } + y ^ { \prime } \right) + p y = 0$ on $\mathbb { R } _ { + } ^ { * }$.