Question

Can somebody help me find the critical points of this function?

Answer 1

\[x ^{1/6}-x ^{-5/6}\]

Answer 2

So I get it down to \[\frac{ 1 }{ 6 }x ^{-5/6}+\frac{ 5 }{ 6 }x ^{-11/6}\]

Answer 3

Have you considered factoring it?

\(x^{1/6}(??)\)

Answer 4

Would it factor out like this @tkhunny? \[x ^{1/6}(1-x ^{-5/6})\]

Answer 5

i think you can factor out a bit more...\[\frac{ 1 }{ 6 }x^{-\frac{ 5 }{ 6 }}\]

Answer 6

So would that be 0?

Answer 7

yeah, but what's left? you get another root

Answer 8

-5?

Answer 9

what, you're not confident?

Answer 10

It said 0 and -5 were wrong @pgpilot326.

Answer 11

\[f \left( x \right)=x^{\frac{ 1 }{ 6 }}-x^{-\frac{ 5 }{ 6 }}\]
so 0 is not in the domain of the origianl function.

Answer 12

\[f'\left( x \right)=\frac{ 1 }{ 6 }x^{-\frac{ 5 }{ 6 }}+\frac{ 5 }{ 6 }x^{-\frac{ 11 }{ 6 }}=\frac{ 1 }{ 6 }x^{-\frac{ 5 }{ 6 }}\left( 1+5x^{-1} \right)\]right?

Answer 13

domain: x > 0

Answer 14

so no critical points, yeah?

Answer 15

Oh...You are right.

Answer 16

have you guys plotted using desmos?

Answer 17

why?

Answer 18

0 is not in the domain of the original function

Answer 19

Thanks man @pgpilot326 !

Answer 20

Well there's some lovely asymptotes at zero but it still does not fit the definition of a critical number being excluded from the domain f .

Answer 21

asymptotic activity that is...

Answer 22

Really?

\(f(x) = x^{1/6} - x^{-5/6} = x^{1/6}\left(1 - x^{-1}\right)\)

It was just one suggestion how you might have proceeded. You should brush up on your algebra before trying to learn the calculus.

Answer 23

good one tk! that was my first step too.
they're learning to run before they can walk.