Extrema and Points of Inflection
Extrema and Points of Inflection
TGM Original
| At a stationary point | Conclusion |
|---|---|
| \(\mathrm{f}^{\prime\prime}(a)>0\) | Local minimum |
| \(\mathrm{f}^{\prime\prime}(a)<0\) | Local maximum |
| \(\mathrm{f}^{\prime\prime}(a)=0\) | Inconclusive; inspect derivative signs |
An inflection point requires a change of concavity. Solving \(\mathrm{f}^{\prime\prime}(a)=0\) supplies candidates, not proof. It may have zero or non-zero gradient.
| Function at the origin | First derivative | Second derivative | Conclusion |
|---|---|---|---|
| \(x^3\) | \(3x^2\) | \(6x\) | Stationary inflection |
| \(x^3+x\) | \(3x^2+1\) | \(6x\) | Non-stationary inflection |
| \(x^4\) | \(4x^3\) | \(12x^2\) | Local minimum; no inflection |
\(y=x^3+x\)
\(y=x^4\)
For \(x^4\), the second derivative is positive on both sides of zero. For \(x^3+x\), it changes from negative to positive while the first derivative at zero is \(1\).
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