Solution
= Solution
Since $f\prime\prime\prime\lt 0$, $f\prime\prime$ decreases from $0$, so $f\prime$ strictly decreases from $1$ and $f(x)\lt x$. If $f\prime$ never vanished, its decreasing positive <limit> would make $f$ stay positive, but $f\prime\prime\lt 0$ bounded away from zero after any fixed point forces $f\prime$ negative eventually. Thus $f$ rises to one maximum and then decreases strictly to $-\infty$, crossing zero exactly once; it is positive before that crossing.