How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = x^3 + 6x^2$ ?

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = x^3 + 6x^2$ ?

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Answer 1 · 2015-12-20T18:06:18

Find when $f'(x)=0$ or $"DNE"$ .

$f'(x)=3x^2+12x=3x(x+4)$

$f'(x)=0$ when $x=-4,0$ .
$f'(x)$ never $"DNE"$ .

Now, use a sign chart with $-4,0$ .

$f'(x)color(white)(xxxxxxxx)-4color(white)(xxxxxxxxxxxxx)0$
$larr------------------rarr$
$color(white)(xxxx)"POSITIVE"color(white)(xxxxx)"NEGATIVE"color(white)(xxxxxx)"POSITIVE"$

$f$ is increasing whenever $f'(x)>0$ .
$f$ is decreasing whenever $f'(x)<0$ .

Thus,

$f$ is increasing on $(-oo,-4)uu(0,+oo)$ .
$f$ is decreasing on $(-4,0)$ .

A relative maximum occurs whenever $f'$ switches from positive to negative.
A relative minimum occurs whenever $f'$ switches from negative to positive.

Thus,

There is a relative maximum when $x=-4$ .
There is a relative minimum when $x=0$ .

graph{x^3+6x^2 [-51.76, 65.27, -14.2, 44.35]}

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = x^3 + 6x^2$ ?

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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 + 6x^2f (x) = x^3 + 6x^2?

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How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for $f (x) = x^3 + 6x^2$ ?