Solution
= Solution
Four differentiations give
$$
f^{(4)}(x)=e^x-ex,\qquad f^{(5)}(x)=e^x-e.
$$
For $0<x<1$, the fifth <derivative> is negative, so the fourth <derivative> is strictly decreasing. Since $f^{(4)}(1)=0$, strict decrease towards that endpoint implies $f^{(4)}(x)>0$ for $x<1$. On the closed <interval>, \b[$0\le f^{(4)}(x)\le1$], with the maximum at $0$ and minimum at $1$.