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Second Derivative Test
Definition: Suppose f(x) is a function of x that is twice differentiable at a stationary point x_0. 1. If f^()(x_0)>0, then f has a local minimum at x_0. 2. If f^()(x_0)<0, then f has a local maximum at x_0. The extremum test gives slightly more general conditions under which a function with f^('')(x_0)=0 is a maximum or minimum. If f(x,y) is a two-dimensional function that has a relative extremum at a point (x_0,y_0) and has continuous partial derivatives at this point, then f_x(x_0,y_0)=0 and f_y(x_0,y_0)=0. The second partial derivatives test classifies the point as a local maximum or relative minimum. Define the second derivative test discriminant as D = f_(xx)f_(yy) - f_(xy)f_(yx) = f_(xx)f_(yy) - f_(xy)^2. Then 1. If D>0 and f_(xx)(x_0,y_0)>0, the point is a relative minimum. 2. If D>0 and f_(xx)(x_0,y_0)<0, the point is a relative maximum. 3. If D<0, the point is a saddle point. 4. If D=0, higher order tests must be used.