`C(x) = x^(1/3) (x + 4)` (a) FInd the intervals of increase or decrease. (b) Find the local maximum and minimum values.

Textbook Question

Chapter 4, 4.3 - Problem 41 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to determine the monotony of the function, hence, you need to find the intervals where f'(x)>0 or f'(x)<0.

You need to find the derivative of the function, using the product rule:

`f'(x) = (x^(1/3))'(x+4) + (x^(1/3))(x+4)'`

`f'(x) = (x+4)/(3x^(2/3)) + (x^(1/3))`

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

`(x+4)/(3x^(2/3)) + (x^(1/3)) = 0`

`x + 4 + (3x^(2/3))(x^(1/3)) = 0`

`x + 4 + 3x = 0 => 4x + 4 = 0 => x = -1`

Hence, f'(x)<0 and the function decreases for `x in (-oo,-1) ` and f'(x)>0, the function increases for `x in (-1,oo)` .

b) Since for x = -1, f'(-1) = 0 and considering the monotony of the function, yields that the function has minimum point at x = -1.

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