I'm reading the page $53$ of this paper and I'm trying understand how the second inequality below is derived
Also since $y = x^4e^{-x}$ has its maximum at $x = \frac{1}{4}$ where $y \leq 1$, we have
$$\frac{d^2f}{dz^2} \leq \frac{c}{p^2} \left( \frac{p}{z} \right)^4 e^{-\frac{p}{z}} \leq \frac{c}{p^2}$$
It seems strange to me because $z \in [0, 4\Lambda]$, it is stated that $p$ is chosen appropriately and $p \geq 4 \Lambda$ by choice, then $p \geq z$.
My doubt is how the second inequality was derived from the function $y = x^4e^{-x}$?