Question

Examine the function for relative extrema and saddle points. $$ f(x, y)=(x+y) e^{1-x^{2}-y^{2}} $$

Short answer

Please carry out the numerical work based on these steps and you will have the relative extrema and saddle points of the given function

Step-by-step solution

Calculate the partial derivatives

We need to start the process by calculating the first partial derivatives of \(f\) with respect to \(x\) and \(y\). This gives us \(f_x\) and \(f_y\), which we'll write in the form \(f_x = \frac{\partial f}{\partial x}\) and \(f_y = \frac{\partial f}{\partial y}\). These derivatives can be obtained using the product and chain rules of differentiation.

Solve the first derivative equations

Next, we set the first derivative equations \(f_x = 0\) and \(f_y = 0\) and solve the system of equations for \(x\) and \(y\). These give us the potential relative extrema and saddle points.

Calculate the second partial derivatives

We calculate the second order partial derivatives of \(f\), which is \(f_{xx}\), \(f_{yy}\) and \(f_{xy}\). These derivatives can be obtained from the first partial derivatives by applying the differentiation rules once more. These second-order derivatives will be required to compute the Hessian determinant.

Find the Hessian determinant

Calculate the determinant of the Hessian matrix. The Hessian matrix, \(H(f)\), in this case, is a 2x2 matrix containing the second order derivatives: \[H(f) = \begin{bmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy} \end{bmatrix}\] As \(f_{xy} = f_{yx}\), the determinant, \(D = f_{xx}f_{yy} - (f_{xy})^2\).

Classify the critical points

Lastly, substitute the potential points into the determinant \(D\). If \(D > 0\) and \(f_{xx} > 0\), it's a local minima. If \(D > 0\) and \(f_{xx} < 0\), then it's a local maxima. If \(D < 0\), it's a saddle point.