Total number of parallel tangents of $f_1(x)=x^2-x+1$ and $f_2(x)=x^3-x^2-2 x+1$ is equal to

None of these

$f'_1\left(x_1\right)=2 x_1-1, f'_2\left(x_2\right)=3 x_2^2-2 x_2-2$

Let tangents drawn to the curves $y=f_1(x)$ and $y=f_2(x)$ at $\left(x_1, f\left(x_1\right)\right)$ and $\left(x_2, f\left(x_2\right)\right)$ are parallel, then $2 x_1-1=3 x_2^2-2 x_2-2$, which is possible for infinite number of order pair $\left(x_1, x_2\right)$.