We have, $f:(a, b) \rightarrow\left[\frac{-\sqrt{3}}{4}, \frac{1}{2}\right]$
$$
\begin{aligned}
& f(x)=\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1+\sin x & 1 \\
1+\cos x & 1 & 1
\end{array}\right| \\
& f(x)=\left|\begin{array}{ccc}
1 & 0 & 0 \\
1 & \sin x & 0 \\
1+\cos x & -\cos x & -\cos x
\end{array}\right| \\
& \quad\left[C_3 \rightarrow C_3-C_1, C_2 \rightarrow C_2-C_1\right] \\
& f(x)=-\sin x \cos x \\
& f(x)=-\frac{1}{2} \sin 2 x \\
& f(x) \in\left[\frac{\sqrt{3}}{4}, \frac{1}{2}\right]
\end{aligned}
$$

$$
\begin{array}{rlrl}
& \therefore & -\frac{\sqrt{3}}{4} & \leq \frac{-1}{2} \sin 2 x \leq \frac{1}{2} \\
\Rightarrow & & -1 & \leq \sin 2 x \leq \frac{\sqrt{3}}{2} \\
\Rightarrow & -\frac{\pi}{4} & \leq x \leq \frac{\pi}{6} \\
& \therefore & a & =-\frac{\pi}{4}, b=\frac{\pi}{6}
\end{array}
$$