How Do You Find a Common Tangent Line to Two Different Functions?

[specone]

I was working on a problem, and in my solution I came across a situation which I will try and state in the following question:

Given two functions, f(x) and g(x), how would you find a line such that the line is tangent to f(x) at some point x=a, and tangent to g(x) at some point x=b, assuming such a line exists?

is this even possible? can you solve it without using some kind of approximation method?

thanks for any help

 

[mathman]

The system of equations you have to solve is:

f'(a)=g'(b)=(g(b)-f(a))/(b-a)

You have two equations in two unknowns (a and b). How difficult it is to solve depends very much of the nature of f(x) and g(x).

 

[specone]

The system of equations you have to solve is:

f'(a)=g'(b)=(g(b)-f(a))/(b-a)

You have two equations in two unknowns (a and b). How difficult it is to solve depends very much of the nature of f(x) and g(x).

thanks man! perfect

 

[fluxions22]

I was working on a problem, and in my solution I came across a situation which I will try and state in the following question:

Given two functions, f(x) and g(x), how would you find a line such that the line is tangent to f(x) at some point x=a, and tangent to g(x) at some point x=b, assuming such a line exists?

is this even possible? can you solve it without using some kind of approximation method?

thanks for any help

it sounds like you would evaluate the slope of the line in question and as long as the slope exists for both f(x) and g(x) you would know that the functions are parallel?