The Hessian Matrix is defined as [math] Hess \, f(x) = \left( \begin{array}{} \frac{∂^2f}{∂x^2} & \frac{∂^2f}{∂y \, ∂x} \\ \frac{∂^2f}{∂x \, ∂y} & \frac{∂^2f}{∂y^2} \\ \end{array} \right) [/math][br][br]For the twice continuously differentiable function [math]f:D\left(\subseteq\mathbf{R^2}\right)\rightarrow\mathbf{R}[/math] with grad f(x[sub]0[/sub],y[sub]0[/sub]) = 0 and symetric matrix [math] Hess = \left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ \end{array} \right) [/math] : [br]i) det Hess > 0 and a[sub]1,1[/sub] > 0 then f has a local minimum at (x[sub]0[/sub],y[sub]0[/sub])[br]ii) det Hess > 0 and a[sub]1,1[/sub] < 0 then f has a local maximum at (x[sub]0[/sub],y[sub]0[/sub])[br]iii) det Hess < 0 then f has a saddle point at (x[sub]0[/sub],y[sub]0[/sub])[br][b][br]Click Init button to start[/b][br][br]You can move the point on the surface to find a maximum, minimum or saddle point.[br][br]Check "auto" if you want the point to move automatically to one local maximum point.[br]Check "curves" to draw curves along gradient and normal to it, and level curve for current point.[br][br]You can enter other functions like:[br]f(x,y) = x*y , f(x,y) = 0.5(x³ + x² - x) - 0.5y² , etc.[br][br](Original idea from Andreas Lindner [url=https://ggbm.at/rVmxKNSE]https://ggbm.at/rVmxKNSE[/url])