Question about finding a function $f(x)$ that fits the following bounds

Question

I am in search of a function $f(x)$ that is infinetely differentiable (that is $f^{(n)}(x) \ne 0$ and that the function is defined when differentiated $n$ times) and an interval $[a,b]$ that will satisfy the following constraints where $n > 0$ and $m>0$: $$ \int_a^be^{-f(x)} \frac{d^n}{dx^n}[e^{f(x)}] dx = 0 $$ $$ \int_a^b \frac{d^n}{dx^n}[e^{-f(x)}] \frac{d^m}{dx^m}[e^{f(x)}] dx = 0 \quad\quad\quad n \ne m $$ $$ \int_a^b \frac{d^n}{dx^n}[e^{f(x)}] \frac{d^n}{dx^n}[e^{-f(x)}] dx = a_n $$ I am looking for a function that is non-trivial and is infinitely differentiable. Either by finding a non-trivial answer or by proving such a function $f(x)$ does or does not exist will satisfy this answer.