Have a problem with a HW question, I understand how to solve these but I am sorta lost on this one.

21) Detemine where the graph of the given function is increasing, decreasing, concave up and concave down. Then sketch (dont worry about this part).

g(x)= 3x^4 - 4x^3 +2

Mono = g'(x)= 12x^3- 12x^2

= 12x^2(x-1) so x would equal 1 ????? But wouldn't x = 0 as well?

Concav = g"(x)= 12x^3-12x^2

= 36x^2 - 24x

= 12x(3x-2) x=2/3

Somewhere I am missing something. The answer states the graph decreases from (-infin,1) increases (1,infin).. concave up (-ifin, 0)u(2/3,infin) and concave down (0,2/3).

I am missing why the zero is in the concativity but not the montonicity and if I plug zero into the Mono I get zero. I have no clue..