The second derivative test allows one to determine the nature of the extrema of a given function after computation.

If f(x) is a function twice differentiable at a critical point x_0, then:

  1. if f''(x)>0, then f has a local minimum at x_0
  2. if f''(x)<0, then f has a local maximum at x_0
  3. if f''(x)=0, then the test is inconclusive

If f(x,y) is a two-dimensional function that has a critical point at (x_0, y_0), then let D = f_{xx}f_{yy} - f_{xy}

  1. If D>0 and f_{xx}(x_0,y_0)>0, then the point is a local minimum
  2. If D>0 and f_{xx}(x_0,y_0)<0, then the point is a local maximum
  3. If D<0, then the point is a saddle point
  4. If D=0, then a higher-order test must be used.

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