Find possible solution for minimization of a functional with free boundary condition

Find a possible solution for the minimization of the functional:
$$J[x]=int_0^1 (tdot{x}+dot{x}^2) , dttag1$$

with $x(0)=1$ and $x(1)$ is a free variable.

I am trying to solve the above, but not sure whether I am going the right way!

My attempt:

Try to find the function $x$, perhaps the line passing from points $x(0)$ and $x(1)$? If $x(0)=1$, and $x(1)= c,cinmathbb{R}$ then $$x(t)= (c-1)t+1. tag2$$

Then replace $(2)$ into the equation $(1)$ and now $$J[x]=int_0^1 (c-1)t+(c-1)^2 , dt$$

Doing the calculations
$$=c^2 -frac{3}{2}c+frac{1}{2}tag3$$

So the problem becomes minimizing $(3)$.

Let $F(c)=c^2 -frac{3}{2}c+frac{1}{2}$. Then $F(c)$ has a critical point at $c=frac{3}{4}$ and it is a minimum for $F(c)$ based on the second order condition. Hence, one possible minimization of the functional is:

$$x(t)= frac{-1}{4}t+1 $$

But not sure if it is a correct approach or I should use Euler-Lagrange like in this example Minimization of functional using Euler-Lagrange