Thread: rational functions question

can someone please help me please?

given the function f(x) = 2x/(x-4) determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through A(5,10) and B(8,4)

the answer is x = 5, x=8; x= 6.5

thanks in advance =)

First, work out the slope of the secant line :


Which gives

Now that you know the slope of the secant line, you can derivate and find a derived point of which has the same slope as the secant line.
The derivate of is :





Now you want a slope of , right ? Therefore, you will want to solve the derivative like this :


Which gives :

Now solve this simple equation :




Factorize this (and by the way, on right-hand-side) :

Easy solve :
So, or

Now you know that the points that have x-coordinate and have a slope of on the first function. Substitute back into the first function to find their y-coordinate :




Let's conclude : when or , the tangent to this point of the function has a slope of , which is equal to the slope of the secant line .
So, the solutions are the points and .