Let $\gamma\in[0,1]$, $M>0$, $n\in\mathbb{Z}^{+}$ and $$ P\leq \exp\left(\frac{M}{5}\right)\exp\left(-n\gamma x + \frac{nx^2}{M-5x}\right),\qquad \text{for }x\in[0, M/5). $$ I want to show that I can reformulate an upper bound that is independent of $x$, by minimising it in $[0,M/5)$. In particular, I want to show that $$ P\leq \exp\left(\frac{M}{5}\right)\exp\left(-n\frac{M\gamma^2}{12}\right). $$ $P$ denotes a probability and can only output values between $0$ and $1$. $P$ is independent of $x$.
This new upper bound works when I check it using some graphing tools. However, I want to derive it analytically from the info I have here. Also, the value $\exp(-nM\gamma^2/12)$ is not the local minimum point of the original bound in $[0,M/5)$. It is just a value large enough for us to justify this uniform bound. Can anyone provide any insights to this question?
This doesnt seem right since for $x = M/5-\epsilon$ the RHS is arbitrarily large.