Extrema and Points of Inflection

**At a stationary point** | **Conclusion** f (a)>0 | Local minimum f (a)<0 | Local maximum f (a)=0 | Inconclusive; inspect derivative signs An inflection point requires a change of concavity. Solving f (a)=0 supplies candidates, not proof. It may have zero or non-zero gradient. **Function at the origin** | **First derivative** | **Second derivative** | **Conclusion** x3 | 3x2 | 6x | Stationary inflection x3+x | 3x2+1 | 6x | Non-stationary inflection x4 | 4x3 | 12x2 | Local minimum; no inflection y=x3+x y=x4 For x4, the second derivative is positive on both sides of zero. For x3+x, it changes from negative to positive while the first derivative at zero is 1.