# 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 $f''(x)>0$, 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?

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

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

1. If $$ff$$ has an extreme value at $$xx$$, then $$pp$$ is even and the form $$f^{(p)}(x)h^{(p)}f^{(p)}(x)h^{(p)}$$ is semidefinite.
2. If there is a constant $$cc$$ such that $$f^{(p)}(x)h^{(p)}\ge c > 0f^{(p)}(x)h^{(p)}\ge c > 0$$ for all $$|h|=1|h|=1$$, then $$ff$$ has a strict local minimum at $$xx$$ and (1) applies.
3. If there is a constant $$cc$$ such that $$f^{(p)}(x)h^{(p)}\le c < 0f^{(p)}(x)h^{(p)}\le c < 0$$ for all $$|h|=1|h|=1$$, then $$ff$$ has a strict local maximum at $$xx$$ 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 $$EE$$ is finite-dimensional. Then $$h\mapsto f^{(p)}(x)h^{(p)}h\mapsto f^{(p)}(x)h^{(p)}$$ has both a minimum and maximum value on the set $$\{h\in E:|h|=1\}\{h\in E:|h|=1\}$$, and:

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

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