Integrals and Inverse Functions

[XJellieBX]

Homework Statement

Set f(x)=\int^{2x}_{1}\sqrt{16 + t^{4}}dt.
A. Show that f has an inverse.
B. Find (f^{-1})'(0).

Homework Equations

(f^{-1})'(x)=1/(f'(f^{-1}(x)))

The Attempt at a Solution

A. f'(x)=\sqrt{16 + t^{4}} >0, so f is always increasing, hence one-to-one. By definition of inverse functions, f would have an inverse.

B. (f^{-1})'(0)=1/(f'(f^{-1}(0)))

I'm not sure where to go from here for part B. Help?

 

[ak123456]

I think maybe you can find the inverse of the function firstly,then subtitute x by 0 ? I am also focusing on part B

 

[Hurkyl]

(Solving f(x) = 0 should be easy...)

You did forget something, though: to have an inverse, a function must be one-to-one and onto. So, for example, you would need to show that \lim_{x \rightarrow +\infty} f(x) = +\infty[/itex], and similarly for -\infty.

 

[XJellieBX]

I think maybe you can find the inverse of the function firstly,then subtitute x by 0 ? I am also focusing on part B

Thanks, I think I've got it. I was overlooking some details.

 

[HallsofIvy]

It is not necessary to actually determine f or f^-1. Since f is defined by an integral, the "fundamental theorem of calculus" gives you its derivative.