The slope of the line [tex]y=2x[/tex] is 2, and any parallel line will have the same slope. Since [tex]f'(x_0)[/tex] gives the tangent line to [tex]y=f(x)[/tex] at [tex]x=x_0[/tex], any tangent line will be parallel whenever
[tex]\arctan(x^3-x)=2[/tex]
But this never happens, since [tex]|\arctan x|\le\dfrac\pi2<2[/tex]
So there will never be a tangent line with slope 2, and thus no tangent line parallel to [tex]y=2x[/tex].
