A function is strictly decreasing on an interval when the derivative is negative.
Given: f(x)=x4−3x3
Finding the derivative using the power rule:
f′(x)=4x3−33x2
f′(x)=4x3−x2
Factoring out common terms:
f′(x)=x2(4x−1)
Setting f′(x)=0:
x2(4x−1)=0
This gives:
x2=0 → x=0
4x−1=0 → x=41
Critical points: x=0 and x=41
Analyzing the sign of f′(x)=x2(4x−1) in each interval:
Note that x2 is always non-negative. The sign of f′(x) depends on (4x−1).
For x<41:
(4x−1)<0, so f′(x)≤0
The function is decreasing.
For x>41:
(4x−1)>0, so f′(x)>0
The function is increasing.
The function is strictly decreasing on (−∞,41).