Question

Locate the stationary points of the function $$ f(x, y)=\left(x^{2}-2 y^{2}\right) \exp \left[-\left(x^{2}+y^{2}\right) / a^{2}\right] $$ where \(a\) is a non-zero constant. Sketch the function along the \(x\) - and \(y\)-axes and hence identify the nature and values of the stationary points.

Short answer

1. Find and solve \(f_x = 0\) and \(f_y = 0\). 2. Use the second derivative test with the Hessian matrix to identify the nature of the stationary points. 3. Sketch the function using these points.

Step-by-step solution

Find the First Partial Derivatives

Compute the first partial derivatives of the function with respect to both variables, x and y. Use: \[ f_x = \frac{\partial f}{\partial x} \quad \text{and} \quad f_y = \frac{\partial f}{\partial y} \]

Set Partial Derivatives to Zero

Set \( f_x = 0 \) and \( f_y = 0 \) and solve the resulting system of equations to find the critical points.

Second Partial Derivatives

Calculate the second partial derivatives \( f_{xx}, f_{yy}, \text{and } f_{xy} \). These will be used for the second derivative test.

Apply the Second Derivative Test

Determine the nature of each stationary point using the second derivative test. Construct the Hessian matrix \( H \), and calculate its determinant \( |H| \).The Hessian determinant is given by: \[ |H| = f_{xx} f_{yy} - (f_{xy})^2 \] Depending on the sign of \( |H| \) and \( f_{xx} \), identify whether each point is a local minimum, local maximum, or a saddle point.

Evaluate the Stationary Points

Substitute the values of stationary points back into \( f(x, y) \) to find their functional values.

Sketch the Function

Draw the function along the x- and y-axes using the identified stationary points to indicate the nature of the function.

Key concepts

Partial Derivatives

Partial derivatives help us understand how a multivariable function changes as we vary one of its input variables while keeping others constant. For a function like \( f(x, y) \), the partial derivative with respect to \( x \) is denoted as \( f_x \) and is computed by differentiating \( f \) with respect to \( x \). Similarly, the partial derivative with respect to \( y \) is denoted as \( f_y \).
In our problem, to find the stationary points of \( f(x, y) = (x^2 - 2y^2) \, e^{-\frac{x^2 + y^2}{a^2}} \), we first need to compute \( f_x \) and \( f_y \). This step helps us identify where the function's slope is zero in all directions because these could be points of interest like maxima, minima, or saddle points.
By setting both \( f_x \) and \( f_y \) to zero, we can solve for \( x \) and \( y \). These solutions give us the 'critical points' that we need.

• Understand that taking partial derivatives is like taking regular derivatives but focusing only on one variable at a time.
• In more complicated functions, use product rule, chain rule, or quotient rule as needed.

Critical Points

Critical points occur where the first partial derivatives of a function are zero. For \( f(x, y) \), we find the critical points by setting \( f_x = 0 \) and \( f_y = 0 \). This gives us a system of equations to solve for \( x \) and \( y \).
These points are where the slope of the function is zero in both directions, meaning it can be a peak, a valley, or a saddle. It's important to correctly solve these equations to get the right critical points.
For the function \( f(x, y) \), solving for \( f_x = 0 \) and \( f_y = 0 \) might lead to multiple solutions which can be real or complex numbers.

• At these points, the function isn’t changing in the direction of x or y.
• Doesn’t necessarily mean we have a maximum or minimum; it could be a saddle point.

Hessian Matrix

The Hessian matrix is essential for analyzing the nature of critical points. It's a square matrix of second-order partial derivatives. For a function \( f(x, y) \), the Hessian matrix \( H \) looks like this:
\ [ H = \begin{pmatrix} f_{xx} & f_{xy} \ f_{xy} & f_{yy} \ \right] \
Here, \( f_{xx} \) is the second partial derivative with respect to \( x \), \( f_{yy} \) is the second partial derivative with respect to \( y \), and \( f_{xy} \) is the mixed partial derivative.
In our problem, computing the Hessian matrix helps us with the second derivative test to determine the nature of the critical points. We will plug the critical points into the Hessian matrix to evaluate it and determine properties like concavity.

• Ensures a systematic approach to checking the concavity or convexity of a function at critical points.
• The determinant of the Hessian matrix helps in deciding if a critical point is a maxima, minima, or saddle point.

Second Derivative Test

The second derivative test uses the Hessian matrix to determine the nature of critical points. After computing the second partial derivatives, we construct the Hessian matrix and calculate its determinant, \( |H| \).
If \( |H| > 0 \) and \( f_{xx} > 0 \), the critical point is a local minimum. If \( |H| > 0 \) and \( f_{xx} < 0 \), it's a local maximum. If \( |H| < 0 \), the critical point is a saddle point where the function neither has a max nor a min but changes direction.
For the given function, we have to compute \( |H| \), substitute the critical points, and determine their nature based on the above conditions.

• Helpful to conclusively categorize the type of stationary points aside from just knowing their location.
• Important for understanding the topography of the function, like hills, valleys, and plains.