3. Given the function
$$
f(x)=\sin \left(\omega x+\frac{\pi}{4}\right)(\omega>0)
$$

has a maximum value but no minimum value in the interval $\left(\frac{\pi}{12}, \frac{\pi}{3}\right)$. Then the range of $\omega$ is . $\qquad$
3. $\left(\frac{3}{4}, 3\right)$.

Since $\frac{\pi}{12}0$, therefore,
$$
\frac{\omega \pi}{12}+\frac{\pi}{4}0$, thus, $k \geqslant 0$.
If $k=0$, then $\frac{3}{4}<\omega<3$;
If $k \geqslant 1$, the system of equations has no solution.
Hence, $\omega \in\left(\frac{3}{4}, 3\right)$.