Expert: Paul Klarreich - 11/18/2009

Question
Please help me find the intervals of f(x) increasing, the interval of f(x) decreasing and local extrema.

Please help me find the intervals of f(x) increasing, the interval of f(x) decreasing and local extrema.

I already found f(0)does not exist and is discontious at 0.

I found first subinterval (-infinite,0, cannot find 2 other subintervals

f(x) = 3 x^(2/7) -  x^(3/7)

Answer
Questioner: Vickkie
Country: United States
Category: Calculus
Private: Yes
Subject: Help Derivatives
Question: Please help me find the intervals of f(x) increasing, the interval of f(x) decreasing and local extrema.


I already found f(0) does not exist

>> yes, it does.

and is discontiNous at 0.

>> No, it isn't.

>> But f'(0) does not exist.


I found first subinterval (-infinite,0) cannot find 2 other subintervals

f(x) = 3 x^(2/7) -  x^(3/7)


Find the derivative:

f'(x) = 6/7 x^(-5/7) - 3/7 x^(-4/7)

Do some work on it:
          6             3
f'(x) = ---------  - ---------
       7 x^(5/7)    7x^(4/7)


          6         3x^(1/7)
f'(x) = ---------  - ---------
       7 x^(5/7)    7x^(5/7)


         6 - 3x^(1/7)
f'(x) = ---------------
          7 x^(5/7)

You have a critical point at  x = 0 (a singular point)  and:

you can set  6 - 3x^(1/7) = 0

6 = 3x^(1/7)

2 = x^(1/7)

x = 2^7 = 128.

That is also a critical point. (A stationary point)

So your intervals are  (-inf, 0),  (0,128), (128,inf)

You can work out the rest.