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 , then:
- if , then f has a local minimum at
- if , then f has a local maximum at
- if , then the test is inconclusive
If f(x,y) is a two-dimensional function that has a critical point at , then let
- If D>0 and , then the point is a local minimum
- If D>0 and , then the point is a local maximum
- If D<0, then the point is a saddle point
- If D=0, then a higher-order test must be used.