Question

How do you find the intervals of increasing, decreasing and concavity for `f(x) = 2x^3 + 3x^2 - 432x`?

Answer 1

Values of `f` increase when `f'(x)>0` and decrease when `f'(x)<0`. Also `f` is concave when `f''(x)<0`.

So:
`f'(x)=2*3x^2+3*2x-432=6x^2+6x-432=`
`=6(x^2+x-72)=6(x+9)(x-8)`
`f'(x)<0 <=> x in (-9;8)`
`f'(x)>0 <=> x in (-oo;-9) or x in (8;+oo)`

`f''(x)=6*2x+6=12x+6=12(x+1/2)`
`f''(x)<0 <=> x in (-oo;-1/2)`