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

1. terenzreignz

Critical points are points where the derivative is zero.

2. terenzreignz

Of course, differentiate first :)

3. eroshea

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

4. terenzreignz

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

5. eroshea

6. 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.

7. eroshea

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

8. 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)

9. eroshea

owh.. i see

10. eroshea

so what next? how will i graph?

11. 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

12. eroshea

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

13. terenzreignz

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

14. eroshea

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

15. terenzreignz

I think it leads nowhere :)

16. eroshea

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

17. terenzreignz

yeah :)

18. eroshea

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

19. terenzreignz

Hang on, I may have made a mistake.

20. terenzreignz

Okay... apparently, 2 was right.

21. terenzreignz

sorry.

22. eroshea

yehey :D so.. what to do next ?

23. 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

24. terenzreignz

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

25. eroshea

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

26. terenzreignz

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

27. terenzreignz

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

28. eroshea

that's ok.. :))

29. terenzreignz

We already have the critical points.

30. eroshea

then how to graph?

31. terenzreignz

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

32. eroshea

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

33. terenzreignz

No problem :)