Question

Consider the function f(x) = -2 x^3 + 36 x^2 - 162 x + 2. There are two critical points, A and B where A < B:
A = and B =
f''(A)=
f''(B)=
Therefore f(x) has a relative at A (MAX or MIN)
and a relative at B (MAX or MIN).

Answer 1

plain old power rule here.

f'(x) = -6x^2 + 72x - 162
= -6(x^2 - 12x + 27)

f"(x) = -12x + 72
= -12(x-6)

So, f' = 0 when x = 3 or 9
f"(3) > 0 so f is concave up (min)
f"(9) < 0 so f is concave down (max)

Note that you can also just rely on what you know about the general shape of cubics to identify the max/min-ness of the extrema.