Expert: Paul Klarreich - 3/16/2006

Question
Hi again, i tried to do this problem, i am just looking for verification that it is correct. If i did something wrong can u please point out what i did?

find the critical numbers of the function:

f(z)=(z+1)/(z^2+z+1)

my answer was: z=0 and z=-2

thanks.


Answer
Hi, chevonne,

Subject:  critical numbers
Question:  Hi again, i tried to do this problem, i am just looking for verification that it is correct. If i did something wrong can u please point out what i did?

find the critical numbers of the function:

f(z)=(z+1)/(z^2+z+1)

my answer was: z=0 and z=-2

thanks.
-------------------------
That's better.  The critical numbers are the zeroes and singularities of f'(z).

       (z^2 + z + 1)(1) - (z + 1)(2z + 1)
f'(z) = ----------------------------------
             (z^2+z+1)^2

       z^2 + z + 1 - [2z^2 + 3z + 1]
f'(z) = -----------------------------
             (z^2+z+1)^2

       z^2 + z + 1 - 2z^2 - 3z - 1
f'(z) = ---------------------------
             (z^2+z+1)^2

       - z^2  - 2z
f'(z) = ----------------
         (z^2+z+1)^2

The zeroes are the zeroes of the numerator (since the denominator is never zero -- you can check that out.) and they are indeed  z = 0 and z = -2