# rational functions question

• Nov 4th 2009, 07:31 PM
mzto
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

• Nov 4th 2009, 08:37 PM
Bacterius
First, work out the slope $m$ of the secant line :

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

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

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

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

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

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

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

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