First verify that $\mathrm{f}^{\prime}(a)=0$. \(\mathrm{f}^{\prime\prime}(a)

Question

First verify that $\mathrm{f}^{\prime}(a)=0$. $\mathrm{f}^{\prime\prime}(a)<0:\ \text{local maximum}\qquad \mathrm{f}^{\prime\prime}(a)>0:\ \text{local minimum}$ If $\mathrm{f}^{\prime\prime}(a)=0$, the test is inconclusive. Use the first-derivative sign test; for inflexion, verify a change in concavity. Example: for $y=x^4$, $\dfrac{\mathrm d^2y}{\mathrm dx^2}=12x^2=0$ at $x=0$, so the test is inconclusive; $\dfrac{\mathrm dy}{\mathrm dx}=4x^3$ changes from negative to positive, so $(0,0)$ is a minimum point.

Accepted Answer

At $\mathrm{f}'(a)=0$: $\mathrm{f}''(a)0$ gives a local minimum, and $\mathrm{f}''(a)=0$ is inconclusive. A stationary inflexion requires a change in concavity.

Second Derivative Test

Second Derivative Test

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