Finding the Extrema of a Function

Date: 10/25/96 at 7:28:28
From: Anonymous
Subject: Extrema

Dear Dr. Math,

I have to analyze the following function:

f(x)=(ln(1+exp(-a-bx-cx^2)))/((x+d)^2)

Questions:

 - If -b/2c>d: is it possible to obtain an analytical expression for 
   the maximum (if it exists) ?

 - Is it possible to calculate when f(d)=f(maximum) ?

I have already determined that without 1/((x+d)^2) I can obtain the 
maximum, x=-b/2c.

I hope that I have been clear.
Thanks and kind regards,

   Henk Huinink

Date: 11/05/96 at 21:07:46
From: Doctor Lorenzo
Subject: Re: Extrema

First of all, there is no global maximum, since as x approaches -d the
function blows up.  So we'll look for local maxima.

Let's think of your function as the product of two (non-negative) 
factors, A = ln (1 + exp (-a-bx-cx^2)) and  B = (x+d)^{-2}.  As you 
already worked out, the factor A is increasing for x < -b/2c and 
decreasing for x> -b/2c.  B, meanwhile, is increasing for x<-d and 
decreasing for x> -d.  The only way to have a local maximum is to have 
one factor increasing while the other is decreasing, so any maximum 
(if it exists) has to occur between -d and -b/2c.  This is true 
regardless of whether -d > -b/2c or -d < -b/2c.  

We treat the cases one at a time.  First assume -d > -b/2c.  At 
x = -b/2c the function is  increasing, as A is stationary and B is 
increasing.  As x approaches -d (from below), the function is 
certainly increasing, and B is blowing up while A isn't going to zero.  
So there are two possibilities:

If c is large enough, you will get one maximum and one mimimum.  
A will decrease rapidly fairly soon after you get away from x=-b/2c, 
but B will increase substantially only later. However, if c is small, 
then A B' will be larger (in magnitude) than A' B throughout the 
interval, and you will have no maxima and no minima.

The case -d < -b/2c is similar. For x slightly greater than -d, the
derivative is sharply negative. For x near -b/2c, the derivative is
somewhat negative. If c is large enough, the derivative will change 
signs twice in the intermediate region, while if c is small the 
derivative will stay negative throughout the region.

I don't see any real hope of finding a closed-form expression for the
location of the extrema, but perhaps somebody more clever than I can 
come up with one.

-Doctor Lorenzo,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/