Analysis of Signs of Derivatives of an Inverse Function

Date: 06/22/2005 at 09:37:35
From: Tommy
Subject: The inverse of a concave function

I want to be able to reach conclusions about the first and second 
order derivatives of an inverse function.  I have a concave function
which I know has f'> 0 and f''< 0.  It is a continuous function and 
can be differentiated as many times as one would like.  The function 
has an inverse, since the Jakobi determinant is not zero.

So now I would like to take the inverse of the function f', and be 
able to conclude what signs the derivaties of this inverse function 
have.  I know that f''< 0 and since f'> 0 and hence cannot take 
negative values, f'''> 0 has to hold.  Now my difficulties arise.  Can 
one conclude anything about the derivatives of the inverse of a 
function (f') with these characteristics?    

Date: 06/23/2005 at 17:55:51
From: Doctor Vogler
Subject: Re: The inverse of a concave function

Hi Tommy,

Thanks for writing to Dr. Math.  Recall that an inverse function is
defined to be a function g such that

  f(g(x)) = x

and

  g(f(x)) = x

for all values x.  So we can take the derivative of the first equation
and get

  f'(g(x))g'(x) = 1,

which means that

  g'(x) = 1/f'(g(x)).

Since f' is always positive, so too g' is always positive.

Then we can take more derivatives of this too:

  g"(x) = [g'(x)]'
        = [1/f'(g(x))]'
        = -[f'(g(x))]'/f'(g(x))^2
        = -[f"(g(x))g'(x)]/f'(g(x))^2
        = -f"(g(x))/f'(g(x))^3.

Now since f' is always positive, and f" is always negative, this means
that g" is always positive.

And so on.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/27/2005 at 06:22:02
From: Tommy
Subject: Thank you (The inverse of a concave function)

Dear Dr. Vogler,

Thank you so much for your reply.  It was the definition of an inverse 
function that had escaped me.  Your help is much appreciated!

Regards, Tommy