Question

Consider the function f(x)=-2x^3+33x^2-108x+2. For this function, there are three important intervals: (-Inf,A], [A,B], [B,Inf) where A and B are the critical points. Find A and B and for each of the important intervals, tell whether f(x) is increasing or decreasing.

Answer 1

Get the derivative with respect to x:
f(x) = -2x^3 + 33x^2 - 108x + 2
f'(x) = -6x^2 + 66x - 108
Let f'(x) = 0 and get the values of x:
0 = -6x^2 + 66x - 108
0 = x^2 - 11x + 18
0 = (x - 2)(x - 9)
x = 2 and x = 9
Therefore, A = 2 and B = 9.
At (-Inf , 2] : decreasing (because f'(x) < 0 at these values of x)
At [2,9] : increasing (because f'(x) > 0 at these values of x)
At [9,+Inf) : decreasing (because f'(x) < 0 at these values of x)

Hope this helps :3