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[parent] relations between Hessian matrix and local extrema (Result)

Let $ x$ be a vector, and let $ H(x)$ be the Hessian for $ f$ at a point $ x$. Let $ f$ have continuous partial derivatives of first and second order in a neighborhood of $ x$. Let $ \nabla f (x)= 0$.

If $ H(x)$ is positive definite, then $ x$ is a strict local minimum for $ f$.

If $ x$ is a local minimum for $ x$, then $ H(x)$ is positive semidefinite.

If $ H(x)$ is negative definite, then $ x$ is a strict local maximum for $ f$.

If $ x$ is a local maximum for $ x$, then $ H(x)$ is negative semidefinite.

If $ H(x)$ is indefinite, $ x$ is a nondegenerate saddle point.

If the case when the dimension of $ x$ is 1 (i.e. $ f: \mathbb{R} \to \mathbb{R}$), this reduces to the Second Derivative Test, which is as follows:

Let the neighborhood of $ x$ be in the domain for $ f$, and let $ f$ have continuous partial derivatives of first and second order. Let $ f'(x) = 0$. If $ f''(x) > 0$, then $ x$ is a strict local minimum. If $ f''(x) < 0$, then $ x$ is a strict local maximum. In the case that $ f''(x)=0$, being $ f'''(x)\neq 0$, $ x$ is said to be an inflexion point (also called turning point). A typical example is $ f(x)=\sin x$, $ f''(x)=-\sin x=0$, $ x=n\pi$, $ n=0, \pm 1, \pm 2, \dots$, $ f'''(x)=-\cos x$, $ f'''(n\pi)=-\cos n\pi=(-1)^{n+1}\neq 0$.



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See Also: extrema, extremum, Hessian form, tests for local extrema in Lagrange multiplier method

Also defines:  second derivative test

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Cross-references: inflexion point, domain, dimension, saddle point, nondegenerate, indefinite, negative semidefinite, local maximum, strict local maximum, positive semidefinite, local minimum, strict local minimum, neighborhood, second order, partial derivatives, continuous, point, Hessian, vector
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This is version 11 of relations between Hessian matrix and local extrema, born on 2002-08-28, modified 2008-01-01.
Object id is 3375, canonical name is RelationsBetweenHessianMatrixAndLocalExtrema.
Accessed 11135 times total.

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AMS MSC: 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)
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