Strongly minimizing curve optimisation with Weierstrass condition No idea where to start on this one:

Find the strongly minimizing curve and value of $J_{min}$ for cases:
  $$\int_1^2 (t^2\dot{x}^2 + 2x^2) dt$$
  where $x(1)=0,x(2)=7$ Using the Weierstrass condition specifically

How is this done? I know not many people do optimization on here, so if anyone knows how to go about this, please atleast give me a link in the right direction.
 A: In general, using the Euler-Lagrange equations we have:$$\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} = 4x - \frac{d}{dt} (2t^2\dot{x}) = 0$$
$$\Rightarrow 2x=2t x'+t^2 x''$$ 
$$ \Rightarrow x(t) = a t + \frac{b}{t^2}$$
And using the given boundary conditions: $a+b = 0,\qquad2a+b/4=7$ yeild:$$x(t) = 4t-\frac{4}{t^2}$$
This is the general solution for a smooth solution. For a solution with corners, you would use the Weierstrass-Erdmann condition. I'm not sure if this is what the problem is referring to or not. But in that case the condtion only states that $x(t)$ has to be continuous at each point (including corners). 
I've experimented a bit. And I can't see that adding corners helps. From what I've seen it only adds to the value of $J_\min$ (which by the way is 140 without corners). For example say you put in a corner at $t=1.5$, then we now have a piecewise function. 
$$x(t)=\left\{ \begin{array}[cc] (a t-\frac{a}{t^2}, & t < 1.5 \\ \frac{152 a}{37 t^2}-\frac{19a t}{37}  -\frac{756}{37 t^2}+\frac{224 t}{37}, & t > 1.5\end{array}\right.$$ Where the mess is due to satisfying the boundary condition and continuity (Weierstrass condition). Notice that we now have a corner for $a \ne 4$, but $\frac{d}{da}J_\min(a) = \frac{7}{37} (114 a-456)$ and thus $J_\min$ is minimized exactly when $a =4 $as we had previously found (without corners).
