Problem 17 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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, \(D = f_{xx}f_{yy} - (f_{xy})^2\).

Step 5: 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.