Find the tangent line(s) to f(x) = (x + 1)/(x − 5)
with slope m = −6

9514 1404 393

Step-by-step explanation:

Each tangent line will be of the form ...

  y = -6x + b

where b = f(x1) +6·x1 for the points f'(x1) = -6.

Taking the derivative, we get ...

  f'(x) = ((x -5)(1) -(x+1)(1))/(x -5)^2 = -6/(x -5)^2

This will have a value of -6 when |x-5| = 1, or x=4 and x=6.

So, the lines are ...

at x=4

  y = -6x + (f(4) +6·4) = -6x +(5/-1)+24

  y = -6x +19

at x=6

  y = -6 +(f(6) +6·6) = -6x +(7/1)+36

  y = -6x +43

72 is the answer to this problem