- #1

Cosmophile

- 111

- 2

## Homework Statement

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Identify the open intervals on which the function ##f(x) = 12x-x^3## is increasing or decreasing

## Homework Equations

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##f(x)=12x-x^3##

##\frac {df}{dx} = 12-3x^2 = -3(x^2 - 4)##

## The Attempt at a Solution

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I'm reading out of two textbooks. One is a Larson/Edwards text, which says to use test values upon intervals and whatnot. However, my other textbook (Serge Lang), as well as the MIT 18.01 lecture videos I am watching, don't use this method and instead utilize inequalities. I prefer this method, but am having a difficult time exercising it, as my high school did very, very little with inequalities, so my experience is minimal. Here is what I've got:

**##f## is increasing when**##f'(x) > 0##

##-3(x^2 - 4) > 0##

##(x^2 - 4) < 0##

##x^2 < 4##

##x < \pm 2##

##x < 2##, ##x< -2## = ##-x > 2##

Here is where I get stuck. I'm not sure how to make sense of this answer. I know that ##f## is increasing on the interval ##(-2, 2)## and decreasing on ##(-\infty, -2)(2, \infty)##, but don't know how to translate my answer to say this. I'm sure I'm either making a simple mistake or am just missing a bit of know-how regarding these types of problems. Any help is greatly appreciated; thanks in advance!