Determine where the function \(f(x)=(x^{2}-1)^{2/3}\) is increasing and where it is decreasing.
The Increasing/Decreasing Function Test states that \(f\) is increasing on intervals where \(f^\prime (x) >0\) and decreasing on intervals where \(f^\prime (x) <0.\) We solve these inequalities by using the numbers \(-1,\) \(0,\) and \(1\) to form four intervals. Then we determine the sign of \(f^\prime\) in each interval, as shown in Table 2. We conclude that \(f\) is increasing on the intervals \((-1,0)\) and \((1,\infty)\) and that \(f\) is decreasing on the intervals \((-\infty ,-1)\) and \((0,1)\).
| Interval | Sign of \(4x\) | Sign of \((x^{2}-1) ^{1/3}\) | Sign of \(f'\left( x \right) = \frac{{4x}}{{3{{\left( {{x^2} - 1} \right)}^{1/3}}}}\) | Conclusion |
|---|---|---|---|---|
| \(( -\infty ,-1)\) | Negative \((-)\) | Positive \((+)\) | Negative \((-)\) | \(f\) is decreasing on \((-\infty,-1)\) |
| \((-1,0)\) | Negative \((-)\) | Negative \((-)\) | Positive \((+)\) | \(f\) is increasing on \((-1,0)\) |
| \((0,1)\) | Positive \((+)\) | Negative \((-)\) | Negative \((-)\) | \(f\) is decreasing on \((0,1)\) |
| \((1,\infty)\) | Positive \((+)\) | Positive \((+)\) | Positive \((+)\) | \(f\) is increasing on \((1,\infty)\) |