3. The function
$$
f(x)=\sqrt{x^{2}+\left(\frac{x^{2}}{4}-2\right)^{2}}+\sqrt{x^{2}+\left(\frac{x^{2}}{4}-1\right)^{2}}
$$

has a minimum value of . $\qquad$
3. 3 .

Notice that the function $f(x)$ can be regarded as the distance from points on the parabola $x^{2}=4 y$ to $(0,2)$ and $(0,1)$.

Since $(0,1)$ is the focus of the parabola, the minimum value is the distance from $(0,2)$ to the directrix of $x^{2}=4 y$, which is $y=-1$, i.e., 3.