7. Find the minimum value of the function $f(x)=\sqrt{\frac{1}{4} x^{4}-2 x^{2}-4 x+13}+\frac{1}{2} x^{2}+\frac{1}{2}$.
7. $f(x)=\sqrt{\left(\frac{x^{2}}{2}-3\right)^{2}+(x-2)^{2}}+\sqrt{\left(\frac{x^{2}}{2}-\frac{1}{2}\right)^{2}+(x-0)^{2}}$, can be regarded as the sum of the distances from a point $(x, \left.\frac{x^{2}}{2}\right)$ on the parabola $y=\frac{1}{2} x^{2}$ to two fixed points $A(2,3), B\left(0, \frac{1}{2}\right)$. Since $B$ is exactly the focus, the distance from $A$ to its directrix $\frac{7}{2}$ is the desired result.