What is the higher-order derivative test in multivariable calculus?

In single-variable calculus, the second-derivative test states that if x is a real number such that f(x)=0, then:

  1. If f, then f has a local minimum at x.
  2. If f”(x)<0, then f has a local maximum at x.
  3. If f''(x)=0, then the text is inconclusive.

But there's no need to despair if the second-derivative test is inconclusive, because there is the higher-order derivative test. It states that if x is a real number such that f'(x)=0, and n is the smallest natural number such that f^{(n)}(x)\neq 0, then:

  1. If n is even and f^{(n)}>0, then f has a local minimum at x.
  2. If n is even and f^{(n)}<0, then f has a local manimum at x.
  3. If n is odd, then f has an inflection point at x.

Similarly, in multivariable calculus the second-derivative test states that if (x,y) is an ordered pair such that \nabla f(x,y) = 0, then:

  1. If D(x,y)>0 and f_{xx}(x,y)>0, then f has a local minimum at (x,y).
  2. If D(x,y)>0 and f_{xx}(x,y)<0, then f has a local maximum at (x,y).
  3. If D(x,y)<0, then f has a saddle point at (x,y).
  4. If D(x,y)=0, then the test is inconclusive.

where D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-(f_{xy}(x,y))^2 is the determinant of the Hessian matrix of f evaluated at (x,y).

My question is, what do you do if this test is inconclusive? What is the analogue of the higher-order derivative test in multivariable calculus?

Answer

This webpage states and proves a version of the higher-order derivative test that applies not only to functions defined on \mathbb{R}^2 or \mathbb{R}^N, but functions defined on arbitrary Banach spaces. First there is this theorem:

Theorem 38 (Higher derivative test). Let A\subseteq E be an open set and let f:A\to\mathbb{R}. Assume that f is (p-1) times continuously differentiable and that D^p f(x) exists for some p\ge 2 and x\in A. Also assume that f'(x),\dots,f^{(p-1)}(x)=0 and f^{(p)}(x)\ne 0. Write h^{(p)} for the p-tuple (h,\dots,h).

  1. If f has an extreme value at x, then p is even and the form f^{(p)}(x)h^{(p)} is semidefinite.
  2. If there is a constant c such that f^{(p)}(x)h^{(p)}\ge c > 0 for all |h|=1, then f has a strict local minimum at x and (1) applies.
  3. If there is a constant c such that f^{(p)}(x)h^{(p)}\le c < 0 for all |h|=1, then f has a strict local maximum at x and (1) applies.

Then there is this corollary for the finite dimensional case, which is what we’re interested in:

Corollary 39 (Higher derivative test, finite-dimensional case). In Theorem 38, further assume that E is finite-dimensional. Then h\mapsto f^{(p)}(x)h^{(p)} has both a minimum and maximum value on the set \{h\in E:|h|=1\}, and:

  1. If the form f^{(p)}(x)h^{(p)} is indefinite, then f does not have an extreme value at x.
  2. If the form f^{(p)}(x)h^{(p)} is positive definite, then f has a strict local minimum at x.
  3. If the form f^{(p)}(x)h^{(p)} is negative definite, then f has a strict local maximum at x.

Here f^{(p)}(x) denotes a tensor containing all the pure and mixed partial derivatives of f of order p, evaluated at x.

Attribution
Source : Link , Question Author : Keshav Srinivasan , Answer Author : Keshav Srinivasan

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