Directional Derivative, Gradient and Jacobian

Extreme Values

Second Derivatives Test

If $f:{\mathbb{R}}^2\to \mathbb{R}$ has continuous second order partial derivatives in a neighbourhood of $(a,b)$ and $\frac{\partial f}{\partial x}(a,b)=\frac{\partial f}{\partial y}(a,b)=0$. Let $D=\text{pddf}{}^{2}f{x}^2(a,b)\text{pddf}{}^{2}f{y}^2(a,b)-{\left(\text{pddf}{}^{2}fx\partial y(a,b)\right)}^2$.

• If $D>0$ and either $\text{pddf}{}^{2}f{x}^2(a,b)>0$ or $\text{pddf}{}^{2}f{y}^2(a,b)>0$, then $f$ has a local minimum at $(a,b)$.
• If $D>0$ and either $\text{pddf}{}^{2}f{x}^2(a,b)<0$ or $\text{pddf}{}^{2}f{y}^2(a,b)<0$, then $f$ has a local maximum at $(a,b)$.
• If $D<0$, then $f$ has a saddle point at $(a,b)$.
• If $D=0$, the test is inconclusive. i.e., we need to consider higher order derivatives to determine.