**Department of Mathematics
Education**

**J. Wilson, EMAT 6680**

** **

Matthew Tanner

EMAT 6680

Write-up #1

June 24, 2003

SELECT ONE PROBLEM TO WRITE UP AND POST TO YOUR WEB
PAGE AS HOMEWORK.

3. Find two ** linear **functions

**h(x) =
f(x).g(x)**

is tangent to each of **f(x)
**and **g(x)** at two distinct points. Discuss and illustrate the method
and the results.

First observe that, because *f(x)* and *g(x)*
are linear functions, they are of the form

_{}

_{}.

Next, suppose the two functions are symmetric about
the Y-axis. They’re my lines. And besides, I posit without proof that we
can accomplish this by linear transformation.
This has the effect that *a _{1} = -a_{2}* and

_{}

_{}

and

_{}.

Note that, since *g(x)* and *f(x)* must
each be tangent to *h(x)* and the derivative of a function is equivalent
to its slope, the values of *x* for which *f(x)* and *g(x) *are
tangent to *h(x)* must respectively satisfy

_{} for *f(x)*

_{} _{} for *g(x).*

By inspection, the values _{} and _{}satisfy the equations.
Arbitrarily selecting *a = 1* gives the values _{}.

Precisely because *h(x)* is the product of *f(x)*
and *g(x)*, the points where *f(x)* and *g(x)* are tangent are
also their respective zero points. So

_{},

_{}.

Note that since the tangent points
and hence the zero points are inversely associated with the slopes of their
respective functions, any value for *a* may be chosen. Click
here for the demo.