Question

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`?

Answer 1

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]}