What interval would this function (inside) be increasing and decreasing?
f(x) = 3x^4 - 4x^3 - 12x^2 + 4
Before looking at derivatives, look at the function.
It's a 4th-degree polynomial.
4 is even so it's either going to enter in QII and exit in QI or enter in QIII and exit in QIV
the lead coefficient is positive so it will come in QII and exit QI
it's going to mainly decrease then increase but we still need to see what happens around the y-axis and if there are any bumps in between the decreasing and increasing.
the derivative is 12x^3-12x^2-24x
where the derivative is negative, the function is decreasing and where it is positive it is increasing.
where it is 0 it is changing
12x^3-12x^2-24x=0
12x(x^2-x-2)=0
12x(x-2)(x+1)=0
x=-1,0,2
you need to look at what the derivative is doing in the following intervals:
(-∞,-1),(-1,0),(0,2) and (2,∞)
If x<-1, 12x is negative, (x-2) is negative) and (x+1) is negative. Neg*Neg*Neg is negative. The function is DEcreasing (matches the picture)
If -1<x<0, 12x is negative, (x-2) is negative, (x+1) is positive. Neg*Neg*Pos is positive. The function is INcreasing
If 0<x<2, 12x is positive, (x-2) is negative, (x+1) is positive. Pos*Neg*Pos is negative. The function is DEcreasing.
If x>2, 12x is positive, (x-2) is positive, (x+1) is positive. Pos*Pos*Pos is positive, The function is INcreasing
Increasing: (-1,0) U (2,∞)
Decreasing: (-∞,-1) U (0,2)