First verify that f (a)=0. f (a)<0: local maximum f (a)>0: local minimum Local maximum at (1,2). Local minimum at (1,-2). If f (a)=0, the second-derivative test is inconclusive. Use the first-derivative sign test. For example, y=(x-1)4 has y (1)=0, but y =4(x-1)3 changes from negative to positive at x=1, so (1,0) is a minimum.

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 second-derivative test is inconclusive. Use the first-derivative sign test. For example, \(y=(x-1)^4\) has \(y^{\prime\prime}(1)=0\), but \(y^{\prime}=4(x-1)^3\) changes from negative to positive at \(x=1\), so \((1,0)\) is a minimum.

Second Derivative Test

Second Derivative Test