SOLUTION: Consider the function y = f(x) ={( x + 1/ x - 2)}^3 (a) Find the derivative of f. (b) Find the equations of the tangent and the normal to the curve y = f(x) at the point (1,-8).

Algebra ->  Expressions-with-variables -> SOLUTION: Consider the function y = f(x) ={( x + 1/ x - 2)}^3 (a) Find the derivative of f. (b) Find the equations of the tangent and the normal to the curve y = f(x) at the point (1,-8).      Log On


   

Question 1032139: Consider the function y = f(x) ={( x + 1/ x - 2)}^3
(a) Find the derivative of f.
(b) Find the equations of the tangent and the normal to the curve y = f(x) at the point (1,-8).
(c) Find all the values of x so that f'(x) < 0.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
(a) y+=+f%28x%29+=+%28+%28x+%2B+1%29%2F%28+x+-+2%29%29%5E3+
==> y' =
= %28-9%28x%2B1%29%5E2%29%2F%28x-2%29%5E4
(b) f'(1) = -36 after substitution into the previous equation for y'.
==> the equation of the tangent at (1, -8) is y%5Bt%5D+%2B+8+=+-36%28x-1%29, or
y%5Bt%5D+=+-36x+%2B+28.
==> the equation of the normal at (1, -8) is y%5Bn%5D+%2B+8+=+%281%2F36%29%28x-1%29, or
y%5Bn%5D+=+%281%2F36%29x+-+289%2F36 after simplification.
(c) f'(x) < 0 for all real values EXCEPT at x = 2, where there is an essential (infinite) discontinuity. (This is because f'(x) = %28-9%28x%2B1%29%5E2%29%2F%28x-2%29%5E4+%3C+0 except at x=2 where it is undefined.)