Problem

Describe two lines, *f(x)* and *g(x)* such that *f*
and *g* are tangent
to the product *h(x) = f(x)g(x)*.

We have to get h(x)=f(x) *g(x).

We know for properties of product that h(x) will intersect with
either f(x) or g(x) when either f(x) or g(x) are equal to zero.
We can say that the intercepts of h(x) are the intercepts of f(x)
and g(x)

We also know that h(x) must have to two real roots.

Let us to discover what are the characteristics of the slops and the intercepts of f(x) and g(x).

Let's start with the general case.

If we set the derivatives of *f* and *h*
equal to each other at the first root, then we should find something
about the constants a, b, c, and d.

(1)

Now let us do the same with the derivatives
of *g* and *h*.

(2)

From equations (1) and (2) sets and substitutions we get

a=-c (3)

This is very important information because now we know the characteristics of the slopes

Now lets to discover what happened with the
intercepts. Let set the derivatives of *f* and *h* equal
to each other at the first root, and substitute (3) to determine
what happened with b and d

b+d=1(4)

So with (3) and (4) we have found the characteristics for slopes and intercepts of f(x) and g(x)

For instance f(x) = -x+2 and g(x)=x-1 satisfy the conditions