(10 Given $f(x)=\frac{x^{4}+k x^{2}+1}{x^{4}+x^{2}+1}(k, x \in \mathbf{R})$. Then the product of the maximum value $f(x)_{\text {max }}$ and the minimum value $f(x)_{\text {min }}$ of $f(x)$ is ( ).
(A) $\frac{4 k-1}{3}$
(B) $k$
(C) $\frac{k+1}{2}$
(D) $\frac{k+2}{3}$
$10 \mathrm{D}$ Hint: $f(x)=1+\frac{(k-1) x^{2}}{x^{4}+x^{2}+1}$, and $x^{4}+1 \geqslant 2 x^{2}$, so
$$
0 \leqslant \frac{x^{2}}{x^{4}+x^{2}+1} \leqslant \frac{1}{3} \text {. }
$$

When $k \geqslant 1$, $f(x)_{\text {max }}=\frac{k+2}{3}, f(x)_{\text {min }}=1$;
When $k<1$, $f(x)_{\text {min }}=\frac{k+2}{3}, f(x)_{\text {max }}=1$.
Therefore, $f(x)_{\max } \cdot f(x)_{\min }=\frac{k+2}{3}$.