How do you find the intervals of increasing and decreasing using the first derivative given #y=(x-1)^(1/3)#?

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Answer 1 · 2016-11-21T14:30:10

Differentiate.

Let #y = u^(1/3)# and #u = x- 1#.

#y' = 1/3u^(-2/3)# and #u' = 1#

#dy/dx = 1/(3u^(2/3))#

#dy/dx= 1/(3(x - 1)^(2/3))#

The derivative will have a critical value of #1#, since it renders the derivative undefined. Check to the left and right of this point.

#dy/dx = 1/(3(3 - 1)^(2/3)) = 1/(3(root(3)(4))) ->" positive: the function is " color(blue) "increasing"#

#dy/dx = 1/(3(-2 - 1)^(2/3)) = 1/(3(root(3)(9))) ->" positive: the function is "color(blue) "increasing"#

So, the function #y = (x- 1)^(1/3)# is uniformly increasing.

Hopefully this helps!