1. ## Interval of increase/decrease

Find the interval of increase/decrease of f = x^(1/7) (x+8)

I started to find f' = x^(1/7) + (x+8) (1/7 x^(-6/7)) But when I try to simplify and solve for x, everything gets strange and I know that I am not doing it right. Can you show me the steps, please? Thanks.

2. Originally Posted by Maziana
Find the interval of increase/decrease of f = x^(1/7) (x+8)

I started to find f' = x^(1/7) + (x+8) (1/7 x^(-6/7)) But when I try to simplify and solve for x, everything gets strange and I know that I am not doing it right. Can you show me the steps, please? Thanks.
Try multiplying through first. I think that will make it a little easier.

$f=x^{8/7} + 8x^{1/7}$

3. f' = x^(8/7) + 8x^(1/7)

8/7 x^(1/7) + 8/7 x^(-6/7)

x^(1/7) = -x^(-6/7)

x = -x^-6

x = -1/x^6

x^7 = -1

x = -1

Thanks.

4. Originally Posted by Maziana
f'' = x^(8/7) + 8x^(1/7)

8/7 x^(1/7) + 8/7 x^(-6/7)

x^(1/7) = -x^(-6/7)

x = -x^-6

x = -1/x^6

x^7 = -1

x = -1

Thanks.
Wait, so the first equation is the second derivative? I thought it was the original function.

Edit: But you are right.
x=-1 is one of the zeroes. The other zero is at x=0. However, if you remember the first step after taking the derivative is finding the domain. In this case x <> 0, because 0 in the denominator would be undefined.

Now just test before and after -1 and before and after 0 to find rise/fall.