**Problem #3: **This problem asked to** **find two
** linear **functions

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

is tangent to each of **f(x)
**and **g(x)** at two distinct points. My two linear equations
that I used and their product are as follows:

Here is the graph:

One finds out upon investigating this problem that the only way this problem will work is if these four things occur:

1. The graphs of** f(x)**
and **h(x) **meet at a specific value of x. Let **x = a.**

2. **f'(a) = h'(a)**

3. The graph of **g(x)**
and** h(x) **meet at a specific value of x. Let **x = b.**

4. **g'(b) = h'(b)**

Because the derivate of **h(x)**
will be equal to the derivate of the linear equations at the given
location in #2 and #4, we know that the graph of **h(x**) is
tangent to the graphs of the linear equations at these two points.

Check the solution:

**An interesting exercise:
**Plug in these equations
and let n vary from -10 to 10. Enjoy!!