5. The minimum value of the function $y=-4 x+3 \sqrt{4 x^{2}+1}$ is
Let $2 x=\tan \theta, \theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then $y=-2 \tan \theta+3 \sec \theta=\frac{3-2 \sin \theta}{\cos \theta} \geqslant \sqrt{5}$, equality holds when $2 \sin \theta+\sqrt{5} \cos \theta=3 \Rightarrow \tan \theta=\frac{2 \sqrt{5}}{5} \Rightarrow x=\frac{\sqrt{5}}{5}$. Therefore, the minimum value of the function $y=-4 x+3 \sqrt{4 x^{2}+1}$ is $\sqrt{5}$.