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]}