(1) The maximum value of the function $f(x)=\sin ^{4} x+\sin x \cos x+\cos ^{4} x$ is
(1) $\frac{9}{8}$ Hint: $f(x)=\sin ^{4} x+\sin x \cos x+\cos ^{4} x$
$$
\begin{array}{l}
=\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cos ^{2} x+\sin x \cos x \\
=1-\frac{1}{2} \sin ^{2} 2 x+\frac{1}{2} \sin 2 x \\
=-\frac{1}{2}\left(\sin 2 x-\frac{1}{2}\right)^{2}+\frac{9}{8} .
\end{array}
$$

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