Integrals and Inverse Functions

1. The problem statement, all variables and given/known data
Set f(x)=[tex]\int^{2x}_{1}[/tex][tex]\sqrt{16 + t^{4}}[/tex]dt.
A. Show that f has an inverse.
B. Find ([tex]f^{-1}[/tex])'(0).


2. Relevant equations
([tex]f^{-1}[/tex])'(x)=1/(f'([tex]f^{-1}[/tex](x)))


3. The attempt at a solution
A. f'(x)=[tex]\sqrt{16 + t^{4}}[/tex] >0, so f is always increasing, hence one-to-one. By definition of inverse functions, f would have an inverse.

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

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

Staff Emeritus
Science Advisor
Gold Member
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(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 [tex]\lim_{x \rightarrow +\infty} f(x) = +\infty[/itex], and similarly for [itex]-\infty[/itex].
 
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

Science Advisor
41,626
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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.
 

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