I would like to find the function $v(x)>0,x\in \mathrm{R^+}$ that minimizes $I$, where $I$ is defined as:
$$ \begin{align} I &= \int_{a}^{b} v(x) \cdot x \exp(-\lambda x) \mathrm{d}x, \quad {\lambda,a} \in \mathrm{R^+} \end{align} $$
subject to the following constraint:
$$ \int_a^b v(x) \mathrm{d}x = c, \quad c \text{ is some positive constant} $$
I've looked at the literature and the closest I could find was the Euler - Lagrange equation, which finds the extremum of:
$$ \begin{align} I &= \int_{a}^{b} F(x, v(x), v'(x)) \mathrm{d}x \end{align} $$
Any hints on whether my problem is well defined and what method I could use to determine $v(x)$?