Question

List the critical numbers of the following function in increasing order:

g(x)=3x^(1/3)-3x^(-2/3)

Answer 1

\[g(x)= 3x ^{1/3}-3x ^{-2/3}\]

Answer 2

Have you ever done derivatives?

Answer 3

yeah i got the derivative. And then i know i have to set it to 0. but then i'm having problems solving for x

Answer 4

Ok, what's your derivative?

Answer 5

\[g'(x)=x ^{-2/3}+2x ^{-5/3}\]

Answer 6

Then when I set it equal to 0 and try to solve, the fraction exponents are messing with me hah

Answer 7

Rewrite this in fractional form:\[g'(x)=\frac{1}{x^{2/3}}+\frac{2}{x^{5/3}}\]Find the common denominator. What do you get.

Answer 8

\[\frac{ x ^{5/3}+2x ^{2/3} }{ x ^{10/9} }\]

Answer 9

sorry denominator should be x^(-7/3)

Answer 10

x^(7/3)! my bad haha

Answer 11

Not exactly, it's just algebra at this point, so I'll help you out. This is what that results in when combined:\[\frac{x+2}{x^{5/3}}\]

Answer 12

Actually you just fixed your error... but when simplified it would amount to my answer anyway.

Answer 13

Oh yeah, cool! i can solve it from there then. Thanks man!

Answer 14

Btw, don't forget. You have a critical point when the derivative is undefined as well. So don't forget to solve:\[x^{5/3}=0\]as well.

Answer 15

That would just be 0 wouldn't it?

Answer 16

so -2 and 0 would be the c.p.'s?

Answer 17

yea

Answer 18

Thanks

Answer 19

np