To check if a function is increasing or decreasing, find its derivative and check if it's positive or negative.
If f′(x)>0, the function is increasing.
If f′(x)<0, the function is decreasing.
Given: f(x)=4−3x+3x2−x3
Using the power rule:
f′(x)=0−3(1)+3(2x)−3x2
f′(x)=−3+6x−3x2
Factoring out the common factor −3:
f′(x)=−3(1−2x+x2)
The expression 1−2x+x2 is a perfect square since (x−1)2=x2−2x+1
Therefore:
f′(x)=−3(x−1)2
The term (x−1)2 is always ≥0 for any value of x.
Therefore:
−3(x−1)2≤0 for all x
This means f′(x)≤0 for all real numbers.
Since f′(x)≤0 for all x∈R, the function is decreasing on R.
Note: Although f′(x)=0 at x=1, this is just one point. The function is still considered decreasing on all of R.
Therefore, the function is decreasing on R.