Theorems from Local extrema

The multivariable Fermat condition

Differentiability is sufficient for the partial derivatives to exist, but is stronger than what this coordinatewise condition needs. The condition is necessary, not a classification: a stationary point may be a local maximum, a local minimum, a saddle, or a point requiring higher-order analysis.

The second-order partial derivative test

At a stationary point, the quadratic Taylor term is

For a symmetric Hessian, this quadratic form is positive definite if it is positive for every nonzero $\mathbf{h}$, negative definite if it is negative for every nonzero $\mathbf{h}$ and indefinite if it takes both positive and negative values. The test uses $D$ and $f_{xx}$ to tell these cases apart.

When $D>0$, the quadratic form has one sign in all non-zero directions and $f_{xx}$ determines whether that sign is positive or negative. When $D<0$, the form changes sign, which is the algebraic signature of a saddle. If $D=0$, higher-order terms or direct comparisons along curves are required.

To see the determinant rule when $D>0$, write $a=f_{xx}(\mathbf{x}_0)$, $b=f_{xy}(\mathbf{x}_0)$ and $c=f_{yy}(\mathbf{x}_0)$. Then $a\ne0$ and the quadratic part satisfies

Both coefficients on the right have the sign of $a$, so small nonzero displacements change $f$ in the same direction.

For intuition, a symmetric Hessian has principal directions described by its eigenvectors. Its eigenvalues give the curvatures in those directions: both positive suggests a local minimum, both negative a local maximum and opposite signs a saddle. In two variables, $D$ is the product of those two eigenvalues, while $f_{xx}$ picks the sign when they agree.