A function is increasing when the derivative is positive: f′(x)>0.
Given f(x)=x4−2x2
Using the power rule:
f′(x)=4x3−4x
Factoring:
f′(x)=4x(x2−1)
f′(x)=4x(x−1)(x+1)
Setting f′(x)=0:
4x(x−1)(x+1)=0
Critical points: x=−1,x=0,x=1
These points divide the number line into 4 intervals.
Testing the sign of f′(x)=4x(x−1)(x+1) in each interval:
| Interval | Test Point | Sign of 4x | Sign of (x-1) | Sign of (x+1) | Sign of f'(x) | Increasing? |
|---|---|---|---|---|---|---|
| x<−1 | x=−2 | - | - | - | - | No |
| −1<x<0 | x=−0.5 | - | - | + | + | Yes |
| 0<x<1 | x=0.5 | + | - | + | - | No |
| x>1 | x=2 | + | + | + | + | Yes |
The derivative is positive on (−1,0) and (1,∞).
The function is increasing on (−1,0)∪(1,∞).