2. For the function $y=f(x)(x \in D)$, if for any $x_{1} \in D$, there exists a unique $x_{2} \in D$ such that
$$
\sqrt{f\left(x_{1}\right) f\left(x_{2}\right)}=M \text {, }
$$

then the function $f(x)$ is said to have a geometric mean of $M$ on $D$.
Given $f(x)=x^{3}-x^{2}+1(x \in[1,2])$. Then the geometric mean of the function $f(x)=x^{3}-x^{2}+1$ on $[1,2]$ is $M=$ $\qquad$
2. $\sqrt{5}$.

Notice that, when $10.$
$$

Thus, $f(x)=x^{3}-x^{2}+1$ is an increasing function on the interval $[1,2]$, with its range being $[1,5]$.
Therefore, according to the definition of the geometric mean of the function $f(x)$, we have $M=\sqrt{5}$.