What is a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$? As I understand it, the Euler-Lagrange equation is a necessary but not a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$. 
If this is so, what is a sufficient condition? 
 A: Let me begin with an analogy to high school calculus. We know that a function $f(x)$ has a stationary point $x_0$ if at that point
$$\left.\frac{df}{dx}\right|_{x_0}=0$$
There are three possibilities: minimum, maximum and saddle. This is called the first derivative test. To test for these conditions, we use the second derivative test. We check
$$\left.\frac{d^2f}{dx^2}\right|_{x_0}$$
If this is positive, $f$ is concave up and thus $x_0$ is a minimum. If it is negative, $x_0$ is concave down and a maximum. If both first and second derivatives are zero at $x_0$, it is a point of inflection, more specifically, a saddle point (a point of inflection which has zero first derivative is a saddle).
Now we come to variational calculus. Path integral quantum mechanics tells us that the action 
$$S[q]=\int L\,dt$$
is stationary along the classical path, a condition which we write as
$$\delta S[q][h]=0$$
where the functional derivative of a functional $G[f]$ is defined as
$$\delta G[f][h]=\left.\frac{d}{d\epsilon}G[f+\epsilon h]\right|_{\epsilon=0}$$
The action principle implies the Euler-Lagrange equations
$$\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial\dot q}=0$$
But these equations are only necessary for extremal solutions, and saddle point solutions are possible.
So suppose we solve the EL equations and are not sure if it is a minimum, maximum or saddle point. We perform the second derivative test for functionals as follows:
The second functional derivative is
$$\delta^2 G[f][h]=\left.\frac{d^2}{d\epsilon^2}G[f+\epsilon h]\right|_{\epsilon=0}$$
so we look at the integral
$$\delta^2 S[q_\text{c}][h]$$
where $q_\text{c}$ is the curve which solves the EL equations. If it is positive, $q_\text{c}$ is a minimum, etc.
You should take a look at this paper, "When action is not least". It discusses the differences between minima, maxima and saddle points in great detail.
