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?

1 Answer
Dec 20, 2015

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]}