How Does the Mean Value Theorem Prove an Inequality Involving Tan^-1?

[r34racer01]

Use the mean value theorem to show that (abs. value of tan^-1 a) < (abs. value a) for all a not equal to 0. And use this inequality to find all solutions of the equation tan^-1 x = x.

I have no idea how to do this.

 

[Mark44]

Use the mean value theorem to show that (abs. value of tan^-1 a) < (abs. value a) for all a not equal to 0. And use this inequality to find all solutions of the equation tan^-1 x = x.

I have no idea how to do this.

Start with what the mean value theorem says, and go from there.

 

[r34racer01]

Start with what the mean value theorem says, and go from there.

Well MVT is, if f is cont. on [a,b] and differentiable on (a,b). Then there exists a number c E (a,b) such that: f '(c) = f(b) - f(a)/b-a

But I don't get how to apply that here.

 

[HallsofIvy]

Well, taking f(x)= tan^{-1[sup(x) would be a start. What is the derivative of tan-1(x)?}