For all $n \in \mathbb{N}^\star$, justify that there exists a unique function $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfying $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$.
For all $n \in \mathbb{N}^\star$, justify that there exists a unique function $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfying $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$.