The core idea of this problem is to find a function that is a stationary point for a certain process, expresed as an integral functional, this means that we need to set the functional derivative of the integral to zero.

$$ \delta S[q|\phi] = \frac{\partial}{\partial \varepsilon} \int_a^b F\circ \Gamma(q+\varepsilon \phi) dt = \int_a^b \frac{\partial}{\partial \varepsilon} F(q+\varepsilon \phi, \dot q+\varepsilon \dot\phi)dt = \int_a^b \frac{\partial F}{\partial q}\phi + \frac{\partial F}{\partial \dot q}\dot\phi dt $$