The Hesse matrix ist given by [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][b]Theorem[br][/b]Let [math]f:D\left(\subseteq\mathbf{R^2}\right)\rightarrow\mathbf{R}[/math] be a twice differentiable function with grad f(x[sub]0[/sub],y[sub]0[/sub]) = 0 and [math] Hess = \left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ \end{array} \right) [/math] be a symmetric matrix then[br][br]i) det Hess > 0 ∧ a[sub]1,1[/sub] > 0 ⇒ Hess is positive definite, i. e. in (x[sub]0[/sub],y[sub]0[/sub]) is a local minimum of f.[br]ii) det Hess > 0 ∧ a[sub]1,1[/sub] < 0 ⇒ Hess is negative definite, i. e. in (x[sub]0[/sub],y[sub]0[/sub]) is a local maximum of f.[br]iii) det Hess < 0 ⇒ Hess is indefinite, i. e. in (x[sub]0[/sub],y[sub]0[/sub]) is a saddle point.[br][br][b]Task[/b][br]Move [color=#0000ff][b]point P[/b][/color] and try to find maxima, minima or saddle points.[br]Use functions such as[br]f(x,y) = x*y , f(x,y) = 0.5(x³ + x² - x) - 0.5y² , f(x,y) = sin(x)*sin(y) etc.[br][i][br]Hint: You can change the properties of the grid in the xy-plane to distance π or π /2 as needed.[/i]