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\frac\pi2<2`

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