Increasing, Decreasing, and Concavity

Question

The f(x) = x^3 - 6x^2 + 12x + 6 has the following intervals: (-oo,5)U(7,oo)

Can someone tell me what interval is this function decreasing, what interval is the concave downward and upward, and what is the limit as x approaches positive and negative infinity??

please help, i've been struggling with this question a lot!

Answer 1

I will solve this problem without taking any derivatives.

x^3 - 6x^2 + 12x + 6 = x^3 - 6x^2 + 12x - 8 + 14 = (x - 2)^3 + 14

The function is always increasing and has an inflection at x = 2, or (2, 14)

It is concave downward for x < 2 and concave upward for x > 2