Question 24
a) Let $a \in \mathbb{R}$ and $f : \mathbb{R} \to \mathbb{R}$ be a continuous function, having antiderivative $F : \mathbb{R} \to \mathbb{R}$, such that $F(x) + a \cdot f(x) \ge 0$, for $x \in \mathbb{R}$, and $\lim_{|x|\to\infty} \frac{F(x)}{e^{|\alpha \cdot x|}} = 0$, for any $\alpha \in \mathbb{R}^*$. Prove that $F(x) \ge 0$, for $x \in \mathbb{R}$.
b) Let $n \in \mathbb{N} \setminus \{0,1\}$, $g = X^n + a_1 X^{n-1} + \dots + a_{n-1} X + a_n \in \mathbb{R}[X]$ a polynomial with all its roots real and $f : \mathbb{R} \to \mathbb{R}$, a polynomial function such that $f(x) + a_1 \cdot f'(x) + a_2 \cdot f^{(2)}(x) + \dots + a_n \cdot f^{(n)}(x) \ge 0$, for all $x \in \mathbb{R}$. Show that $f(x) \ge 0$, for $x \in \mathbb{R}$.
b) Let $n \in \mathbb{N} \setminus \{0,1\}$, $g = X^n + a_1 X^{n-1} + \dots + a_{n-1} X + a_n \in \mathbb{R}[X]$ a polynomial with all its roots real and $f : \mathbb{R} \to \mathbb{R}$, a polynomial function such that $f(x) + a_1 \cdot f'(x) + a_2 \cdot f^{(2)}(x) + \dots + a_n \cdot f^{(n)}(x) \ge 0$, for all $x \in \mathbb{R}$. Show that $f(x) \ge 0$, for $x \in \mathbb{R}$.