3. $a, b, c \in \mathbf{R}^{+}, f(x)=\sqrt{x^{2}+a}+\sqrt{(c-x)^{2}+b}$ The minimum value is
3. $\sqrt{c^{2}+(\sqrt{a}+\sqrt{b})^{2}}$.
$f(x)=\sqrt{(x-0)^{2}+(0+\sqrt{a})^{2}}+\sqrt{(x-c)^{2}+(0-\sqrt{b})^{2}}$, which can be seen as the sum of the distances from a moving point $(x, 0)$ on the $x$-axis to two fixed points $A(0,-\sqrt{a})$ and $B(c, \sqrt{b})$, its minimum value is $|A B|=\sqrt{c^{2}+(\sqrt{a}+\sqrt{b})^{2}}$.