Question

find the critical numbers for :

Answer 1

abs (x^2-1)

Answer 2

what would you define as a critical number?

Answer 3

if its about derivatives:

the derivative of abs(u) = u' u/abs(u)

Answer 4

you want zeros and undefineds

Answer 5

that makes f prime (x) = 0

Answer 6

yeah

Answer 7

0, and -+1

Answer 8

I know that , but how ?

Answer 9

how what?

Answer 10

how did you find these critical numbers

Answer 11

\[|x^2-1|'=2x\frac{x^2-1}{|x^2-1|}\]
set it equal to zero, and since its a fraction, a 0 denominator is undefined

Answer 12

x=0, -1, 1

Answer 13

2 conditions for critical values:

f' = 0
f' is undefined

people often forget about the second part

Answer 14

ok so you took the numerator and set it equal to zero , then got x=1 and x=-1 ? since we can simplify it to (x-1)(x+1) ?

Answer 15

and we got a zero from 2x

Answer 16

in this case, thats a good way to see it.

but actaully, i got a zero from 2x, and i got the undefined parts from the factors of x^2-1

Answer 17

0/0 is undefined

Answer 18

i know , but does it matter if I took the numerator in this case ? , if the numerator was x then yes I would look at a way to get it undefined instead of a 0.

Answer 19

thanks by the way

Answer 20

it does not matter in this case since the top and bottom are the same relative function. the absolute value parts acts like a switch that changes from 1 to -1 and back to 1 again. the only place that it is not defined at is when the top and bottom are 0

Answer 21

http://www.wolframalpha.com/input/?i=y%3D%28x%5E2-1%29%2F%7Cx%5E2-1%7C

Answer 22

thank you so much.

Answer 23

youre welcome