## FANDOM

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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 $D = f_{xx}f_{yy} - f_{xy}^2$

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.