3. Given functions defined on $\mathbf{R}$
$$
f(x)=x^{2} \text { and } g(x)=2 x+2 m \text {. }
$$

If the minimum value of $F(x)=f(g(x))-g(f(x))$ is $\frac{1}{4}$, then $m=$ $\qquad$ .
3. $-\frac{1}{4}$.

Notice,
$$
\begin{array}{l}
F(x)=f(g(x))-g(f(x)) \\
=(2 x+2 m)^{2}-\left(2 x^{2}+2 m\right) \\
=2 x^{2}+8 m x+4 m^{2}-2 m \\
=2(x+2 m)^{2}-4 m^{2}-2 m .
\end{array}
$$

Then $-4 m^{2}-2 m=\frac{1}{4} \Rightarrow m=-\frac{1}{4}$.