Tangents to Functions
by
Susan Sexton
Find two linear functions f(x) and g(x) such that their product
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.
I first tried graphing two different functions and their product.
Looking at the result some
modifications needed to be made . . .
First one of the lines needed
to have a negative slope this would create a different product function.
This appears to be better but
the linear functions again need to be modified . . .
Hmmm . . .
Maybe some algebra (and
calculus) may help.
I will first generalize my
two functions:
f(x)=ax+b and g(x)=cx+d
so h(x)=(ax+b)(cx+d)
By distributing h(x) new form
is:
h(x) = acx2 + adx
+ bcx + bd
h(x) = acx2 +
(ad+bc)x + bd
Finding the derivative should
help in finding the points tangent to h(x) at any given x.
h¢(x)=2acx + ad +bc
Since the derivative is the
slope of the line tangent to the function at x, I need to find x-values. Therefore I will find the roots of f(x)
and g(x) which are x1 = -b/a and x2 = -d/c respectively.
By substituting the x values
into the derivative this will yield the slope of the line tangent to h(x) at x.
h¢(x1)= 2acx1 + ad +bc and h¢(x2)=2acx2+ ad +bc
h¢(-b/a)=2ac(-b/a) + ad +bc h¢(-d/c)=2ac(-d/c) +ad +bc
=-2bc + ad + bc =-2ad + ad + bc
=ad – bc =-ad + bc
Setting these equal to the slopes of
the given functions, f(x) and g(x) we get:
a = ad – bc and c
= -ad + bc
This means that the slopes of the
tangent functions are the same but with opposite signs.
So now I adjust my original functions to reflect this:
This looks a lot better! Now to work on the y-intercepts.
Back to the algebra (and calculus) .
. .
Finding that
a = ad – bc and c
= -ad + bc
lead to the conclusion that a = -c.
I need to find a relationship between
b and d (the two y-intercepts).
Using substitution I get:
a = ad – b(-a)
1 = d + b
So the sum of the y-intercepts is 1.
Let me again work on the original
linear functions:
That looks pretty good, but let me
zoom in a little to see if those lines are indeed tangent.
I think that confirms it!
So this may work for any two linear
functions . . .
If f(x) = 3x + 5 then what are the
functions g(x) and h(x) and their respective graphs?
If using strictly algebra (no
calculus involved) is there another approach?