At a stationary point $x=a$, $\mathrm{f}^{\prime}(a)=0$. Check the derivative on both sides. Bef

Question

At a stationary point $x=a$, $\mathrm{f}^{\prime}(a)=0$. Check the derivative on both sides. Before aAt aAfter aConclusion +0−Local maximum −0+Local minimum +0+Neither −0−Neither If the derivative keeps its sign, check for a change in concavity before concluding that the point is an inflexion. Example: for $y=x^3$, $\dfrac{\mathrm dy}{\mathrm dx}=3x^2$ is positive on both sides of $x=0$, so $(0,0)$ is neither a maximum nor a minimum: it is a stationary point of inflexion.

Accepted Answer

Maximum: $\mathrm{f}'$ changes $+$ to $-$. Minimum: $\mathrm{f}'$ changes $-$ to $+$. Stationary inflexion: $\mathrm{f}'(a)=0$, $\mathrm{f}'$ keeps the same sign, and concavity changes.

First Derivative Test

First Derivative Test

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Before a	After a	Conclusion
+	−	Local maximum
−	+	Local minimum
+	+	Neither
−	−	Neither