`f(x) = ln(x^4 + 27)` (a) FInd the intervals of increase or decrease. (b) Find the local maximum and minimum values.

Textbook Question

Chapter 4, 4.3 - Problem 42 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the monotony of the function, hence, you need to remember that the function increases if f'(x)>0 and the function decreases if f'(x)<0.

You need to evaluate the first derivative of the function:

`f'(x) = (ln(x^4+27))' `

`f'(x) = (1/(x^4+27))*(x^4+27)'`

`f'(x) = (4x^3)/(x^4+27)`

You need to set f'(x) = 0:

`(4x^3)/(x^4+27) = 0`

`4x^3 = 0 => x = 0`

You need to notice that f'(x)>0 for `x in (0,+oo)` and f'(x)<0 for `x in (-oo,0), ` hence, the function increases for `x in (0,+oo)` and it decreases for `x in (-oo,0).`

b) The local maximum and minimum values are those x values for f'(x) = 0. From previous point a) yields that f'(x) = 0 for x = 0 and the function decreases as x approaches to 0, from the left, and then it increases.

Hence, the function has only a minimum point at x = 0, and the point is (0, ln27).

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