1. Function
$$
f(x)=\sin ^{4} x+\sin x \cdot \cos x+\cos ^{4} x
$$

The maximum value is $\qquad$.
1. $\frac{9}{8}$.
Notice that,
$$
\begin{array}{l}
f(x)=\sin ^{4} x+\sin x \cdot \cos x+\cos ^{4} x \\
=\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cdot \cos ^{2} x+ \\
\quad \sin x \cdot \cos x \\
=1-\frac{1}{2} \sin ^{2} 2 x+\frac{1}{2} \sin 2 x \\
=-\frac{1}{2}\left(\sin 2 x-\frac{1}{2}\right)^{2}+\frac{9}{8} .
\end{array}
$$

Therefore, when $\sin 2 x=\frac{1}{2}$, $f(x)$ reaches its maximum value $\frac{9}{8}$.