Question

Given the function f (x) = 3x^4 – 6x^3 + 3x^2 . Find (if any)
(i) the relative maximum and relative minimum
(ii) the interval(s) where f(x) is increasing and decreasing.
(iii) the interval(s) where f(x) is concave up and concave down

Answer 1

Come on. By this time you should know all about 1st and 2nd derivatives and what they tell you.

f has max/min where f' = 0 and f'' ≠ 0
f is increasing where f' > 0
f is concave up where f'' > 0

So, knowing that

f' = 12x^3 - 18x^2 + 6x
= 6x(2x^2 - 3x + 1)
= 6x(2x-1)(x-1)
and
f'' = 36x^2 - 36x + 6
= 6(6x^2 - 6x + 1)

that should get you set up.