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

- anonymous

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

- chestercat

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- terenzreignz

Critical points are points where the derivative is zero.

- terenzreignz

Of course, differentiate first :)

- anonymous

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

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## More answers

- terenzreignz

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

- anonymous

is my answer correct?

- terenzreignz

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.

- anonymous

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

- terenzreignz

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

- anonymous

owh.. i see

- anonymous

so what next? how will i graph?

- terenzreignz

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

- anonymous

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

- terenzreignz

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

- anonymous

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

- terenzreignz

I think it leads nowhere :)

- anonymous

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

- terenzreignz

yeah :)

- anonymous

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

- terenzreignz

Hang on, I may have made a mistake.

- terenzreignz

Okay... apparently, 2 was right.

- terenzreignz

sorry.

- anonymous

yehey :D
so.. what to do next ?

- terenzreignz

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

- terenzreignz

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

- anonymous

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

- terenzreignz

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

- terenzreignz

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

- anonymous

that's ok.. :))

- terenzreignz

We already have the critical points.

- anonymous

then how to graph?

- terenzreignz

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

- anonymous

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

- terenzreignz

No problem :)

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