How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = \frac{x^{3}}{x^{2} - 4}$ $f(x)=(x^3)/(x^2-4)$ ?

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = \frac{x^{3}}{x^{2} - 4}$ $f(x)=(x^3)/(x^2-4)$ ?

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Answer 1 · 2017-04-14T10:36:32

The intervals of increasing are $x \in (- \infty, - \sqrt{12}) \cup (\sqrt{12}, + \infty)$ $x in (-oo,-sqrt12) uu(sqrt12, +oo)$
The intervals of decreasing are $x \in (- \sqrt{12}, - 2) \cup (- 2, + 2) \cup (2, \sqrt{12})$ $x in (-sqrt12,-2)uu(-2,+2)uu(2,sqrt12)$

Explanation:

We calculate the first derivative and construct a sign chart.

We need

$(u/v)'=(u'v-uv')/(v^2)$

$f(x)=x^3/(x^2-4)$

$u=x^3$ , $=>$ , $u'=3x^2$

$v=x^2-4$ , $=>$ , $v'=2x$

$f'(x)=(3x^2(x^2-4)-2x*x^3)/(x^2-4)^2$

$=(3x^4-12x^2-2x^4)/(x^2-4)^2$

$=(x^4-12x^2)/(x^2-4)^2$

$=(x^2(x^2-12))/(x^2-4)^2$

$f'(x)=(x^2(x+sqrt12)(x-sqrt12))/((x+2)^2(x-2)^2)$

We construct the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaaaaa)$ $-oo$ $color(white)(aaaa)$ $-sqrt12$ $color(white)(aaa)$ $-2$ $color(white)(aaa)$ $0$ $color(white)(aaa)$ $2$ $color(white)(aaa)$ $sqrt12$ $color(white)(aaa)$ $+oo$

$color(white)(aaaa)$ $x+sqrt12$ $color(white)(aaaaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(a)$ $+$ $color(white)(aa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x+2$ $color(white)(aaaaaaaaaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(a)$ $+$ $color(white)(aa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-2$ $color(white)(aaaaaaaaaaa)$ $+$ $color(white)(aaaa)$ $+$ $color(white)(aaa)$ $+$ $color(white)(a)$ $+$ $color(white)(aa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-sqrt12$ $color(white)(aaaaaaaa)$ $-$ $color(white)(aaaa)$ $-$ $color(white)(aaa)$ $-$ $color(white)(a)$ $-$ $color(white)(aa)$ $-$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f'(x)$ $color(white)(aaaaaaaaaaa)$ $+$ $color(white)(aaaa)$ $-$ $color(white)(aaa)$ $-$ $color(white)(a)$ $-$ $color(white)(aa)$ $-$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaaaaa)$ $↗$ $color(white)(aaaa)$ $↘$ $color(white)(aaa)$ $↘$ $color(white)(a)$ $↘$ $color(white)(aa)$ $↘$ $color(white)(aaaa)$ $↗$

The relative maximum is when $x=-sqrt12$

The relative minimum is when $x= sqrt12$

graph{x^3/(x^2-4) [-14.24, 14.24, -7.12, 7.12]}

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = \frac{x^{3}}{x^{2} - 4}$ $f(x)=(x^3)/(x^2-4)$ ?

1 Answer

Explanation:

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for f(x)=(x^3)/(x^2-4)f(x)=x3x2−4f(x)=(x^3)/(x^2-4)?

1 Answer

Explanation:

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = \frac{x^{3}}{x^{2} - 4}$ $f(x)=(x^3)/(x^2-4)$ ?