`f(x) = (x + 2)^(2/3)` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

Textbook Question

Chapter 3, 3.3 - Problem 29 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the open intervals on which the function is increasing or decreasing, hence, you need to find where the derivative is positive or negative, so, you need to evaluate the first derivative of the function, such that:

`f'(x) = ((x+2)^(2/3))' => f'(x) = 2/(3*root(3)(x+2))`

You need to solve for x the equation f'(x) = 0:

`2/3*root(3)(x+2)) = 0`

You need to notice that there are no values for x that verify the equation `2/3*root(3)(x+2)) = 0`

You need to notice that `f'(x)<0` on intervals `(-oo,-2)` and `f'(x)>0` on `(-2,+oo).`

Hence, the function increases for `x in (-2,+oo)` and the function decreases for `x in (-oo,-2).`

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