
Finding Critical Numbers
Find the critical numbers of the function and describe the behavior of f at these numbers.
$\displaystyle f(x) = x^6(x  2)^5$
I took the derivative using product rule
$\displaystyle f'(x) = 6x^5 (x2)^5 + x^6 5(x2)^4$
Then i simplified and set $\displaystyle f'(x) = 0$
$\displaystyle f'(x) = x^5(x2)^4 [6(x2) + 5x] = 0$
$\displaystyle f'(x) = x^5(x2)^4 [6x12 + 5x] = 0$
$\displaystyle f'(x) = x^5(x2)^4 (x12) = 0$
and the three points i got were
$\displaystyle x=0$ $\displaystyle x=2$ and $\displaystyle x=12$
i know 0 is right, but 2 and 12 is wrong.
what am i doing wrong?

When finding f'(x)=0,
6x  12 + 5x is actually 11x  12. This root gives you x = 12/11.
The critical points are x = {0, 12/11, 2}.