Pre-calculus Function Question

Date: 2/8/96 at 14:51:2
From: Dan Belter
Subject: Pre-Calc Question

Our class was doing work with exponents and logarithms today, and we
came across this function: f(x)=3De^(1/x) 

We were wondering if this function had a local minimum.

Thanks,

-Dan Belter

Date: 8/3/96 at 9:36:22
From: Doctor Jerry
Subject: Re: Pre-Calc Question

I'm not certain whether D is a constant and, if so, whether it is 
positive or negative.  Maybe it means derivative?

If I assume that D is a positive constant and f is defined for all x 
other than 0, then

1. for small positive x, like x=0.001, f is large.
2. for large positive x, like x=1000, f is a little larger than 3D.
3. for small negative x, like x=-1000, f is a little smaller than 3D.
4. for large negative x, like x=-0.001, f is a little larger than 0.

So, as x comes in from negative infinity, the graph decreases from 3D 
and approaches 0 as x approaches 0 from the left.  Since f does not 
take the value 0, it has no local minimum for x<0.  As x continues 
past 0 and becomes large, the graph just decreases from infinity 
towards 3D, but never take on this value.  So, again, f has no local 
minimum (of 3D).

So, f has no local minimum, unless D=0, in which case, f has a local 
minimum for all x not equal to 0.

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