Multivariable Taylor & Hessian Test

Second-order Taylor, the Hessian, and the discriminant test.

The idea

The second-derivative test in two variables. Let $f(x,y)$ have continuous second partial derivatives near a critical point $a$, meaning $\nabla f(a) = 0$, and set $D = f_{xx}(a)\,f_{yy}(a) - f_{xy}(a)^{2}.$ If $D > 0$ and $f_{xx}(a) > 0$, then $a$ is a local minimum; if $D > 0$ and $f_{xx}(a) < 0$, a local maximum; if $D < 0$, a saddle point. If $D = 0$ the test decides nothing, and each of the three outcomes can occur.

The test comes from the second-order Taylor expansion: for such an $f$, $f(a+h) = f(a) + \nabla f(a)\cdot h + \tfrac{1}{2}\,h^{T}H(a)\,h + o(\lVert h \rVert^{2}),$ where the Hessian $H(a)$ is the matrix of second partial derivatives. At a critical point the linear term vanishes, so for small $h$ the sign of $f(a+h) - f(a)$ is the sign of the quadratic form $h^{T}H(a)\,h$. If that form is positive for every $h \neq 0$, then $f$ increases in every direction away from $a$, and $a$ is a local minimum; if it is negative for every $h \neq 0$, a local maximum; if it takes both signs, $f$ rises along one direction and falls along another, which is a saddle. In two variables $D$ is the determinant of $H(a)$, and its sign together with the sign of $f_{xx}(a)$ classifies the form.

Ways to work on it

• Walkthrough. Expand a function to second order and use the Hessian to classify its critical points.
• Practice. Find the discriminant or classify a quadratic's critical point.
• Hardest. Classify the critical points of a non-quadratic surface.

Not sure where to start? Take the ten-question placement test.