Problem 17 - 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

To verify whether a point is a local minimum, maximum, or saddle point, we first find the first order derivatives. Solve these for zero to get critical points then evaluate the second order derivatives at these points. Combined they make the Hessian determinant, whose sign will classify the critical points into relative extrema and saddle points.

Step 1: Evaluate first order partial derivatives

The partial derivatives of \(f(x, y)=(x+y) e^{1-x^{2}-y^{2}}\) with respect to \(x\) and \(y\) are: \[ f_x=e^{1-x^{2}-y^{2}}(1-2x^{2}-y)+ (x+y) e^{1-x^{2}-y^{2}} (-2x) \] \[ f_y=e^{1-x^{2}-y^{2}}(1-x^{2}-2y^{2})+ (x+y) e^{1-x^{2}-y^{2}} (-2y) \]

Step 2: Find critical points

Any point where \(f_x = 0\) and \(f_y = 0\) are simultaneous is a critical point. Solving these two equations will give us the critical points.

Step 3: Compute second order partial derivatives

Evaluate second order partial derivatives \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), \(f_{yy}\).

Step 4: Classify critical points

Use the second partial derivatives to calculate Hessian determinant: \(D= f_{xx}*f_{yy} - f_{xy}*f_{yx}\). If \(D > 0\) and \(f_{xx}>0\), the point is a relative minimum. If \(D > 0\) and \(f_{xx}<0\), the point is a relative maximum. If \(D < 0\), the point is a saddle.

Key Concepts

Critical Points in Calculus

Critical points are pivotal in calculus when analyzing a multivariable function like \( f(x, y)=(x+y)e^{1-x^2-y^2} \). They represent the locations on a graph where the function's gradient is zero or where the partial derivatives do not exist. To find critical points, we calculate the function's first-order partial derivatives and determine where they simultaneously equal zero.

For instance, by setting the first-order partial derivatives \( f_x \) and \( f_y \) of our example function to zero, we are effectively searching for coordinate pairs \( (x, y) \) that might mark spots where relative extrema or saddle points occur on the function's surface. These points are where the function changes behavior, and identifying them is a crucial step in function analysis in multivariable calculus.

Second Order Partial Derivatives

Second order partial derivatives are derivatives taken twice with respect to one or more variables, indicating how the rate of change itself changes. In our function \( f(x, y)=(x+y)e^{1-x^2-y^2} \), after finding the critical points, we delve into second order partial derivatives such as \( f_{xx} \), \( f_{xy} \), \( f_{yx} \), and \( f_{yy} \).

These derivatives give us insight into the curvature of the function's graph at the critical points. For example, \( f_{xx} \) and \( f_{yy} \) describe the concavity in each of the x- and y-directions, respectively, while \( f_{xy} \), which should be equal to \( f_{yx} \) based on Clairaut's theorem, represents the curvature in the direction of the slant or mixed partial derivative.

Hessian Determinant

In the context of multivariable calculus, the Hessian determinant is a key indicator used to classify critical points. It is obtained from the matrix of second order partial derivatives, called the Hessian matrix. For our function, the determinant is calculated as \( D = f_{xx}*f_{yy} - (f_{xy})^2 \).

The value of the Hessian determinant informs us of the nature of a critical point. A positive determinant suggests that the point is a local extremum — a minimum if \( f_{xx} > 0 \) and a maximum if \( f_{xx} < 0 \). Conversely, a negative determinant indicates that the critical point is a saddle point. The Hessian thus plays a crucial role in optimizing functions across multiple dimensions.

Multivariable Calculus

Multivariable calculus extends single-variable calculus concepts to functions of several variables, like \( f(x, y) \) in our example. This field allows us to explore more complex real-world systems where outcomes depend on multiple inputs. In essence, while derivative and integral calculus in one dimension gives us rates of change and total accumulation along a single line, multivariable calculus does so along surfaces and within volumes in higher dimensions.

Understanding the behavior of functions within this framework involves tools like gradient vectors, directional derivatives, multiple integrals, and the various theorems interlinking these concepts, such as Green's, Stokes', and the Divergence Theorem. Such extensive analysis aids in predicting and controlling dynamic systems in fields as varied as physics, engineering, and economics.