1. ## rational functions question

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

First, work out the slope $\displaystyle m$ of the secant line :

$\displaystyle m = (Y_a - Y_b) / (X_a - X_b)$
Which gives $\displaystyle m = -2$

Now that you know the slope of the secant line, you can derivate $\displaystyle f(x)$ and find a derived point of $\displaystyle f(x)$ which has the same slope as the secant line.
The derivate of $\displaystyle f(x)$ is :

$\displaystyle f'(x) = [2(x - 4) - 2x] / (x - 4)^2$
$\displaystyle f'(x) = (2x - 8 - 2x) / (x - 4)^2$
$\displaystyle f'(x) = (-8) / (x -4)^2$

Now you want a slope of $\displaystyle -2$, right ? Therefore, you will want to solve the derivative like this :

$\displaystyle f'(x) = -2$
Which gives :
$\displaystyle (-8) / (x - 4)^2 = -2$
Now solve this simple equation :
$\displaystyle -8 = -2[(x - 4)^2]$
$\displaystyle -8 = -2(x^2 - 8x + 16)$
$\displaystyle -8 = -2x^2 + 16x - 32$
$\displaystyle 0 = -2x^2 + 16x - 24$
Factorize this (and by the way, $\displaystyle 0$ on right-hand-side) :
$\displaystyle -(2x - 12)(x - 2) = 0$
Easy solve : $\displaystyle S = (2 ; 6)$
So, $\displaystyle x = 2$ or $\displaystyle x = 6$

Now you know that the points that have x-coordinate $\displaystyle 2$ and $\displaystyle 6$ have a slope of $\displaystyle -2$ on the first function. Substitute back into the first function to find their y-coordinate :

$\displaystyle f(2) = 4 / (2 - 4) = 4 / (-2) = -2$
$\displaystyle f(6) = 12 / (6 - 4) = 12 / 2 = 6$

Let's conclude : when $\displaystyle x = 2$ or $\displaystyle x = 6$, the tangent to this point of the function has a slope of $\displaystyle -2$, which is equal to the slope of the secant line $\displaystyle (AB)$.
So, the solutions are the points $\displaystyle (2 ; -2)$ and $\displaystyle (6 ; 6)$.