Find where the function f(x)=3x^4-4x^3-12x^2+5 is increasing and decreasing?

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f(x)=3x^4-4x^3-12x^2+5

First we need to find the critical points in which the function changes behavior (increasing or decreasing)

To determine the critical points, we need to find f'(x)

f'(x)=12x^3-12x^2-24x = 12x(x^2-x-2)=12x(x-2)(x+1)

The critical points are 0, 2, -1

Then we have 4 intervals where the function change behavior

(-inf,-1), (-1,0), (0,2), (2,inf)

1. when x<-1 ==> f'(x) >0 ==> f is decreasing

2. when -1<x<0 ==> f'(x) <0 ==> f is increasing

2. when 0<x<2 ==> f'(x) < 0 ==> f is decreasing

4. when x>2 ==> f'(x)>0 ==> f is increasing

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