I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{equation} \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x, \end{aligned} \end{equation} where $f(\kappa,x)$ is nonnegative in the interval of integration, and has a minimum at some $\kappa_x = \kappa(x)$ (i.e. the minimum of $f(\kappa,x)$ is a function of the integration variable $x$).
The problem is that I can't find a way how to rigorously show that some specific value $\kappa$ minimizes the integral $I$ (but not necessarily the integrand or $f(k,x)$). I have plotted $I$ versus $\kappa$, and the minimum exists -- it is above 0.5, however this must be proved.
I tried using Euler's equation (without the differential part as I don't have the derivative $\kappa'$), but the answer it again provides the optimal function $\kappa(x)$, not a single value. My current thought is to find the average $\bar{\kappa} = \mathbb{E}_x\left\{k(x)\right\}$, but this seems to be a guesswork.
Does anyone have any suggestions on how to find the value $\kappa$ (not function $\kappa(x)$) to minimize $I$?
A bit more information about $f(\kappa,x)$. The function $f(\kappa,x)$ looks rather complex: \begin{equation} \begin{aligned} f(\kappa,x) = \frac{\kappa (1-\kappa)}{T - \kappa x} \mathrm{exp}\left(-A\frac{T - \kappa x}{\kappa (1-\kappa)} - B \frac{T^2-xT}{T - \kappa x} \right), \end{aligned} \end{equation} where $A,B$ are positive constants. Note that for $x = 0$ the optimal $\kappa = 0.5$, for $x>0$ I believe $\kappa>0.5$.