Question

Use a computer algebra system to graph the surface and locate any relative extrema and saddle points. $$ z=\left(x^{2}+4 y^{2}\right) e^{1-x^{2}-y^{2}} $$

Short answer

The solution involves calculating first and second order partial derivatives, finding critical points and classifying them as relative extrema or saddle points. Finally, all points are plotted on a graph using a computer algebra system.

Step-by-step solution

Function Analysis

Analyze and understand the mathematical function \( z=\left(x^{2}+4 y^{2}\right) e^{1-x^{2}-y^{2}} \). This is a 3-dimensional function which is a graph of a surface in variables x and y.

Compute First-Order Partial Derivatives

To find extrema points, we need to compute the first order partial derivatives of the function with respect to x and y. Keep one variable constant while differentiating with respect to the other.

Set First-Order Derivatives Equal to Zero

Find critical points of the function by setting the first order partial derivatives equal to zero. Solving these equations will give the x and y values of the critical points.

Compute Second-Order Partial Derivatives

Compute second-order partial derivatives of the function. There are three: the second partial derivative with respect to x, respect to y and a mixed second partial derivative with respect to x then y, or vice versa. These are needed to classify the nature of the critical points.

Classify Critical Points

Substitute the critical points obtained in step 3 into the second order partial derivatives. The sign of the second order partial derivatives will determine the nature of the critical points. If the second derivative is positive, then the function has a relative minimum at that critical point. If negative, it has a relative maximum. If the second derivative test is inconclusive, i.e., the discriminant is zero, then the test gives no information, and the point is called a saddle point.

Use a Computer Algebra System to Graph

Finally, use the computer algebra system to graph the function. While graphing ensure that the x and y coordinates of the critical points located are clearly marked and visible. This will visually confirm the classification of the points as relative extrema or saddle points.