I am trying to solve an unconstrained maximisation problem. The objective function is quadratic in the (single) choice variable. Annoyingly, the choice variable must be an integer. Is there some general theorem that tells me that the solution to this problem is approximately the same (within $\pm 1$) as the solution to the simpler problem obtained by ignoring the integer constraint?
To provide the details, my problem is \begin{equation} \max_{b \in \mathbb{X}} (v - b)\left(1 - p + \frac{pb}{x+1}\right)^{n-1}. \end{equation} where $\mathbb{X} = \{0, 1, ..., x$} for some $x \in \mathbb{N}$, $p \in [0, 1]$, $v \in \mathbb{X}$ and $n \geq 2$. If I simplify the problem by pretending that I could choose any $b\in [0, x]$, not just integer $b$, I obtain the 'solution’: \begin{equation} b^* = \max\left\{\left( \frac{n - 1}{n} \right)v - \left(\frac{1 - p}{p}\right)\frac{x}{n}, \hspace{0.3em}0\right\} \end{equation} Since my objective function is quadratic in the choice variable $b$, and so single peaked around $b^*$, it seems 'obvious' that the actual solution must be close (within $\pm 1$) to the 'pretend' solution $b^*$. I guess there must be a general theorem which assures this?
Many thanks in advance for any ideas or pointers!