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Finding the Extrema of a Function

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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
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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/
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Associated Topics:
High School Calculus

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