The Second Derivatives Test for Functions of Several Variables

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Recall from The Second Derivatives Test for Functions of Two Variables page that if $$z = f(x, y)$$ is a two variable real-valued function whose partial derivatives are continuous on some disk $\mathcal D$ centered at $$(a, b)$$ and if we define $D(a, b) = \frac{\partial^2}{\partial x^2} f(a, b) \frac{\partial^2}{\partial y^2} f(a, b) - \left [ \frac{\partial^2}{\partial y \partial x} f(a, b) \right ]^2$ then:

If $$D(a, b) > 0$$ and $\frac{\partial^2}{\partial x^2} f(a, b) > 0$ then $$f$$ attains a local minimum at $$(a, b)$$ .

If $$D(a, b) > 0$$ and $\frac{\partial^2}{\partial x^2} f(a, b) < 0$ then $$f$$ attains a local maximum at $$(a, b)$$ .

If $$D(a, b) < 0$$ then $$f$$ attains a saddle point at $$(a, b)$$ .

If $$D(a, b) = 0$$ then this test is inconclusive.

We will see that this second derivatives test can be extended for a function $$f$$ of more than two variables. Let's now derive a general version of the second derivatives test for a real-valued function $z = f(\mathbf{x})$ of $$n$$ variables (where $\mathbf{x} = (x_1, x_2, ..., x_n)$ ). It is important to first review the Definite, Semi-Definite and Indefinite Matrices and Hessian Matrices pages first.

Theorem 1 (Second Derivatives Test for Functions of Several Variables): Let

z = f(\mathbf{x}) = f(x_1, x_2, ..., x_n)

be an

$n$

variable real-valued function, and let

\mathbf{a} = (a_1, a_2, ..., a_n)

be a critical point of

$f$

. Let

\mathcal H ( \mathbf{x} ) = \begin{bmatrix} f_{11} (\mathbf{x}) & f_{12} (\mathbf{x}) & \cdots & f_{1n} (\mathbf{x})\\ f_{21} (\mathbf{x}) & f_{22} (\mathbf{x}) & \cdots & f_{2n} (\mathbf{x})\\ \vdots & \vdots & \ddots & \vdots \\ f_{n1} (\mathbf{x}) & f_{n2} (\mathbf{x}) & \cdots & f_{nn} (\mathbf{x}) \end{bmatrix}

be the Hessian Matrix of

$f$

. Suppose that all second partial derivatives of

$f$

are continuous around some neighbourhood of

\mathbf{a}

. Then:
a) If

\mathcal H ( \mathbf{a} )

is positive definite, then

$f$

attains a local minimum value at

\mathbf{a}

.
b) If

\mathcal H ( \mathbf{a} )

is negative definite, then

$f$

attains a local maximum value at

\mathbf{a}

.
c) If

\mathcal H ( \mathbf{a} )

is indefinite, then

$f$

attains a saddle point at

\mathbf{a}

.
d) If

\mathcal H ( \mathbf{a} )

is not positive definite, not negative definite, and not indefinite, then this test is inconclusive.

Note that the second derivatives test for functions of two variables is included in Theorem 1 above as the case when $$n = 2$$ , since if $$z = f(x, y)$$ is a two variable function with continuous second partial derivatives in some neighbourhood of the critical point $(a, b) \in D(f)$ then the Hessian Matrix of $$f$$ at $$(a, b)$$ is $\mathcal H (a, b) = \begin{bmatrix} \frac{\partial^2}{\partial x^2} f(a, b) & \frac{\partial ^2}{\partial y \partial x} f(a, b)\\ \frac{\partial ^2}{\partial x \partial y} f(a, b) & \frac{\partial ^2}{\partial y^2} f(a, b) \end{bmatrix}$ . We see that since the second partial derivatives of $$f$$ are continuous in some neighbourhood of $$(a, b)$$ , then $$H(a, b)$$ is positive definite if both $\begin{vmatrix} \frac{\partial^2}{\partial x^2} f(a, b) & \frac{\partial ^2}{\partial y \partial x} f(a, b)\\ \frac{\partial ^2}{\partial x \partial y} f(a, b) & \frac{\partial ^2}{\partial y^2} f(a, b) \end{vmatrix} = \frac{\partial^2}{\partial x^2} f(a, b) \frac{\partial^2}{\partial y^2} f(a, b) - \left [ \frac{\partial^2}{\partial y \partial x} f(a, b) \right ]^2 > 0$ and $\frac{\partial^2}{\partial x^2} > 0$ and hence $$f$$ attains a local minimum. The other conclusions of the second derivatives test for a two variable function are left for the reader to verify.

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The Second Derivatives Test for Functions of Several Variables