Question

Find the critical points. x^1/3 - x^-2/3

I take the derivatives of each side and I get:

1/3x^-2/3 - (-2/3x^-5/3)
1/3x^(-2/3) + 2/3x^(-5/3)
and since I have a negative exponent I can make it a fraction and the negative goes away, so then I get

1/(3x^(2/3)) + 2/(3x^(5/3)) = 0

I am not sure how to continue from here

Answer 1

Ok ok it's a little tricky from here.
You can factor, but it's confusing to do so with negative exponents :)

Answer 2

\[\Large\rm \frac{1}{3}x^{-2/3}+\frac{2}{3}x^{-5/3}=0\]Pull out the most negative exponent from each term, and also the 1/3,\[\Large\rm \frac{1}{3}x^{-5/3}\left(x^{3/3}+2\right)=0\]

Answer 3

You're dividing the x^(-5/3) out of each term.

To the first term's exponent is having this rule applied to it:\[\Large\rm -\frac{2}{3}-\left(-\frac{5}{3}\right)\]Subtracting exponents since we're dividing terms, yes?

Answer 4

If that's too confusing, then there might be another way to do it.
Like with where you have it setup as fractions.

Answer 5

thank you. I am a little confused. if I continue from where I have it set up, can I somehow add a 1 to the bottom part of the first fraction, x^2/3, to make it x^5/3?

Answer 6

Get a common denominator?
Certainly!
That might actually be an easier approach.

Answer 7

\[\Large\rm \frac{1}{3x^{2/3}}+\frac{2}{3x^{5/3}}\quad=\quad\frac{x}{3x^{5/3}}+\frac{2}{3x^{5/3}}\]

Answer 8

Multiplying the first fraction by x (which is x^(3/3) ) will get the denominator you're looking for.

Answer 9

so I multiply the top and bottom of the first fraction by x?

Answer 10

yes

Answer 11

oh I see! so the power of 1 from the x is added to the power of x^2/3 making it x^5/3?

Answer 12

Yes :) 1 + 2/3 = 5/3

Answer 13

awesome thank you!! can I ask why we multiplied x/x instead of just adding 1 to the power?

Answer 14

Well recall that in the land of math, you can't just simply add stuff willy nilly.
You have to keep things balanced.

Multiplying by x/x doesn't cause any problem for us since x/x is equivalent to 1. If we chose to add 1 to the exponent of the first fraction, that would effectively be the same as multiplying ONLY THE BOTTOM by x, which would be a problem! :O

Answer 15

ohh ok I see!! thank you for the help!

Answer 16

You can bring the fractions together from that point since they have the same denominator.
\[\Large\rm =\frac{x+2}{3x^{5/3}}\]
Shouldn't be too bad to deal with after that.

np! c:

Answer 17

and my critical point is -2! thanks!