5. The function
$$
y=-4 x+3 \sqrt{4 x^{2}+1}
$$

has a minimum value of $\qquad$ .
5. $\sqrt{5}$.

Notice,
$$
\begin{array}{l}
y>-4 x+3 \sqrt{4 x^{2}}=-4 x+|6 x| \geqslant 0 . \\
\text { By }(y+4 x)^{2}=9\left(4 x^{2}+1\right) \\
\Rightarrow 20 x^{2}-8 x y+\left(9-y^{2}\right)=0 \\
\Rightarrow \Delta=64 y^{2}-80\left(9-y^{2}\right) \geqslant 0 \\
\Rightarrow y^{2} \geqslant 5 .
\end{array}
$$

Since $y>0$, then $y \geqslant \sqrt{5}$, which is achieved when $x=\frac{1}{\sqrt{5}}$.