Let $x^{*}(t)$ be the curve which minimizes the functional

$$J(x)=\int_{0}^{1}\left[x^{2}(t)+\dot{x}^{2}(t)\right] d t$$

satisfying $x(0)=0, x(1)=1$. Then the value of $x^{*}\left(\frac{1}{2}\right)$ is

1. $\frac{\sqrt{e}}{1+e}$

2. $\frac{2 \sqrt{e}}{1+e}$

3. $\frac{\sqrt{e}}{1+2 e}$

4. $\frac{2 \sqrt{e}}{1+2 e}$