10. Given
$$
f(x)=\frac{x^{4}+k x^{2}+1}{x^{4}+x^{2}+1}(k, x \in \mathbf{R}) \text {. }
$$

Then the product of the maximum and minimum values of $f(x)$ is
(A) $\frac{4 k-1}{3}$
(B) $k$
(C) $\frac{k+1}{2}$
(D) $\frac{k+2}{3}$
10. D.

Notice that, $f(x)=1+\frac{(k-1) x^{2}}{x^{4}+x^{2}+1}$.
And $x^{4}+1 \geqslant 2 x^{2}$, then $0 \leqslant \frac{x^{2}}{x^{4}+x^{2}+1} \leqslant \frac{1}{3}$.
When $k \geqslant 1$,
$$
f(x)_{\max }=\frac{k+2}{3}, f(x)_{\min }=1 \text {; }
$$

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

Therefore, $f(x)_{\max } f(x)_{\min }=\frac{k+2}{3}$.