I'm learning how to use Calculus of Variations by applying the process to a problem for research. I'm fairly certain a multiplicity of questions such as these have been asked; to help those like myself, I'll try to be as explicit as possible. In that regard, suppose we have the functional
$$ \phi = \int_{0}^{t} F(\mathbf{q}(\tau),\mathbf{x}) d \tau, $$
wherein the boldface denotes vectors as usual. If we presuppose that the functional, $\phi$ is extremized at $\mathbf{q} = \mathbf{g}$ and $\mathbf{x} = \mathbf{p}$, then we can re-define
$$ \mathbf{q} = \mathbf{g} + \epsilon \mathbf{f}(\tau), \quad \mathbf{x} = \mathbf{p} + \delta\mathbf{d}, $$
for small $\epsilon$ and $\delta$ where $\mathbf{f}$ and $\mathbf{d}$ are arbitrary vectors (I say arbitrary as opposed to the normal qualifier that they must vanish on the boundary and be differentiable at least once given that the function F only depends on $\mathbf{q}$ and not $\dot{\mathbf{q}}$ which eradicates the need for integration by parts later on; is that valid?). With this redefinition, the functional evaluates to a function of two variables
$$ \Phi(\epsilon,\delta) = \phi[\mathbf{g} + \epsilon \mathbf{f}(\tau),\mathbf{p} + \delta\mathbf{d}], $$
where $\Phi$ is extremized at $\epsilon = \delta = 0$. At that extrema then, the following two conditions must hold:
$$ \frac{\partial \Phi}{\partial \epsilon}\rvert_{\epsilon = \delta = 0} = \int_{0}^{t} \frac{\partial F}{\partial \epsilon}\rvert_{\epsilon = \delta = 0} d \tau = 0,$$
and
$$ \frac{\partial \Phi}{\partial \delta}\rvert_{\epsilon = \delta = 0} = \int_{0}^{t} \frac{\partial F}{\partial \delta}\rvert_{\epsilon = \delta = 0} d \tau = 0.$$
Simplifying results in the following:
$$ \int_{0}^{t} \bigg( \frac{\partial F}{\partial \mathbf{q}} \mathbf{f} \bigg) \rvert_{\epsilon = \delta = 0} d \tau = 0, \quad \int_{0}^{t} \bigg( \frac{\partial F}{\partial \mathbf{x}} \mathbf{d} \bigg) \rvert_{\epsilon = \delta = 0} d \tau = 0.$$
Given that $\mathbf{f}$ and $\mathbf{d}$ are arbitrary, the fundamental lemma of calculus of variations holds, and we can conclude that the vectors $\mathbf{g},\mathbf{p}$ that extremize the functional satisfy
$$ \frac{\partial F}{\partial \mathbf{q}}\rvert_{\epsilon = \delta = 0} = 0, \quad \frac{\partial F}{\partial \mathbf{x}}\rvert_{\epsilon = \delta = 0} = 0, $$
subject to whatever boundary conditions are enforced. Is the thought process until the above answer correct? Also, is it correct to say that the premise behind the process concerns assuming an extrema exists for a functional to exploit a variation about that extrema and transform the functional to a function of n-variables, for which we can easily ascribe equations to identify the assumed extrema?