Question

If f'(x)=tan^-1(x^3-x) at how many points is the tangent line to the graph of y=f(x) parallel to the line y=2x

Answer 1

The slope of the line $y=2x$ is 2, and any parallel line will have the same slope. Since $f'(x_0)$ gives the tangent line to $y=f(x)$ at $x=x_0$, any tangent line will be parallel whenever

$\arctan(x^3-x)=2$

But this never happens, since $|\arctan x|\le\dfrac\pi2<2$

So there will never be a tangent line with slope 2, and thus no tangent line parallel to $y=2x$.