Hessian matrix (second derivative test)

The Hessian matrix of a scalar function of several variables $f: \R^n \to \R$ describes the local curvature of that function. By taking the determinant of the Hessian matrix at a critical point we can test whether that point is a local maximum, minimum, or saddle point. The Hessian matrix is defined as:

\mathrm{Hessian}(f) = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_n} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \\ \end{pmatrix}.

In the case of a function of two variables:

\mathrm{Hessian}(f(x, y)) = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}.

Note

$f_x$ is one way to write the partial derivative of the function $f$ with respect to $x$ . It means the same thing as $\frac{\partial f}{\partial x}$ .

Interpreting the Hessian determinant in two variables

The determinant of the Hessian matrix of a function $f$ at a critical point $\mathbf p$ can tell us whether $\mathbf p$ is a local minimum, maximum, or saddle point. Let the Hessian determinant at a point $x_1, y_1$ be $H = \mathrm{Det}(\mathrm{Hessian}(f)|_{(x_1, y_1)})$ :

If $H>0$ and both $f_x(x_1, y_1) > 0$ and $f_y(x_1, y_1) > 0$ , then $(x_1, y_1)$ is a local minimum.
If $H>0$ and both $f_x(x_1, y_1) < 0$ and $f_y(x_1, y_1) < 0$ , then $(x_1, y_1)$ is a local maximum.
If $H<0$ then $(x_1, y_1)$ is a saddle point.
If $H=0$ then the test is inconclusive.