Question

find all the critical point of f(x) = 3x^(5/3) - 15x^(2/3)

Answer 1

Critical points are points where the derivative is zero.

Answer 2

Of course, differentiate first :)

Answer 3

ok.. i got x=0, x=2

Answer 4

let's see....
\[\Large \frac{d}{dx}\left(3x^{\frac53}-15x^{\frac23}\right)=5x^{\frac23}-10x^{-\frac13}\]

Answer 5

is my answer correct?

Answer 6

Try 0.
the derivative doesn't exist, because you have...
\[\Large 5x^{\frac23}-10x^{-\frac13}=5x^{\frac23}-\frac1{10x^{\frac13}}\]
an x in the denmoinator.

Answer 7

so that will be disregarded right? so i will have 2 for the value of x

Answer 8

so yes, that is a critical point (sorry, I forgot to mention that points where the derivative does not exist are also critical points)

Answer 9

owh.. i see

Answer 10

so what next? how will i graph?

Answer 11

Hang on, I'm not sure about the 2 yet. Let's try equating the derivative to zero.
\[\Large0=5x^{\frac23}-\frac1{10x^{\frac13}}\]\[\Large5x^{\frac23}=\frac1{10x^{\frac13}}\]\[\Large5x^{\frac23}\cdot 10x^{\frac13}=1\]\[\Large50x=1\]
Are you sure it's x = 2? :P

Answer 12

can you factor out 5x^(1/3) from 5x^(2/3)−10x^(−1/3)?

Answer 13

You can... but why would you do that?

Answer 14

what will be the answer if you would factor out that one? 5x^(-1/3)
i got a little confused

Answer 15

I think it leads nowhere :)

Answer 16

owh.. lol.. so x = 1/50?

Answer 17

yeah :)

Answer 18

what is the answer if i would take out 5x^(-1/3) from 5x^(2/3)
i really am confused on what will be the outcome

Answer 19

Hang on, I may have made a mistake.

Answer 20

Okay... apparently, 2 was right.

Answer 21

sorry.

Answer 22

yehey :D

so.. what to do next ?

Answer 23

just like that? You're not going to ask me where my error was? :/
You didn't spot it yet... means you may well commit the same... come on, a little challenge :P

Answer 24

\[\Large0=5x^{\frac23}-\frac1{10x^{\frac13}}\]
I should not have put 10 in the denominator. Big mistake.
Correct would be
\[\Large0=5x^{\frac23}-\frac{10}{x^{\frac13}}\color{green}\checkmark \]

Answer 25

you shouldn't include 10 in the denominator.. it should be on the numerator :D

Answer 26

yeah that :)
sorry about that.
I might be getting drowsy.

Answer 27

And as far as I know... we're done.

Answer 28

that's ok.. :))

Answer 29

We already have the critical points.

Answer 30

then how to graph?

Answer 31

whoops... you'll need to consult one of the big guys for that :3
I'm rather terrible at graphing, you see...

Answer 32

owh.. that'll be okay then.. thanks for the help! :D

Answer 33

No problem :)