# Integrals and Inverse Functions

#### XJellieBX

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

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

3. 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

Staff Emeritus
Gold Member
(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 $-\infty$.

#### 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.

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving