Problem 19 - Calculus with Concepts in Calculus

Question

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=e^{x y} $$

Step-by-Step Solution

Answer

The critical point \((0, 0)\) is a saddle point.

Step 1: Find the First Partial Derivatives

To find the critical points, we first need to calculate the first partial derivatives of the function \( f(x, y) = e^{xy} \) with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is given by \( f_x = \frac{\partial}{\partial x}(e^{xy}) = ye^{xy} \). Similarly, the partial derivative with respect to \( y \) is \( f_y = \frac{\partial}{\partial y}(e^{xy}) = xe^{xy} \).

Step 2: Set First Partial Derivatives to Zero

Next, we set each of the first partial derivatives equal to zero to find the critical points. Solving \( f_x = ye^{xy} = 0 \) gives us \( y = 0 \) (since \( e^{xy} eq 0 \) for any real \( x \) and \( y \)). Solving \( f_y = xe^{xy} = 0 \) gives us \( x = 0 \). Therefore, the critical point is \((0, 0)\).

Step 3: Calculate the Second Partial Derivatives

Next, we calculate the second partial derivatives: \( f_{xx} = \frac{\partial^2}{\partial x^2}(e^{xy}) = y^2e^{xy} \), \( f_{yy} = \frac{\partial^2}{\partial y^2}(e^{xy}) = x^2e^{xy} \), and \( f_{xy} = \frac{\partial^2}{\partial x \partial y}(e^{xy}) = e^{xy} + xye^{xy} \).

Step 4: Use the Second Derivative Test

The Second Derivative Test uses the determinant of the Hessian matrix at the critical point \((x_0, y_0)\), \(D = f_{xx}f_{yy} - (f_{xy})^2\). At \((0, 0)\), we find \( f_{xx} = f_{yy} = 0\) and \(f_{xy} = 1\), resulting in \(D = 0 \cdot 0 - 1^2 = -1\). Because \(D < 0\), the test indicates that \((0, 0)\) is a saddle point.

Step 5: Conclusion

Based on calculations using the first and second partial derivatives, the only critical point found is \((0, 0)\), and it is a saddle point because the Hessian determinant is negative.

Key Concepts

Partial Derivatives

Partial derivatives are fundamental in multivariable calculus, helping in the analysis of functions of several variables. They measure the rate at which the function changes as one variable changes while keeping the others constant. In the context of finding critical points for a function, the first step is to compute these derivatives for each variable.
For the given function,

1. To find the partial derivative with respect to x, treat y as a constant and differentiate: \( f_x = \frac{\partial}{\partial x}(e^{xy}) = ye^{xy} \).
2. Similarly, for y, treat x as a constant: \( f_y = \frac{\partial}{\partial y}(e^{xy}) = xe^{xy} \).

This gives us the tools to identify points where both partial derivatives equal zero. These are potential critical points of the function.

Second Derivative Test

The Second Derivative Test is a useful method to determine the nature of critical points found by setting all first partial derivatives to zero. It helps us to decide whether these points are relative maxima, minima, or saddle points. The test involves calculating the second partial derivatives and evaluating a determinant.

Consider the steps:

• Compute the second partial derivatives: \( f_{xx}, f_{yy}, \text{and } f_{xy} \).
• Construct the Hessian determinant: \( D = f_{xx}f_{yy} - (f_{xy})^2 \).

If the determinant \( D \):- is positive and \( f_{xx} > 0 \), you have a local minimum.- is positive and \( f_{xx} < 0 \), you have a local maximum.- is negative, the point is a saddle point.
This test helps in analyzing the concavity of the function at critical points.

Hessian Matrix

The Hessian matrix plays a crucial role in the evaluation of critical points in multivariable calculus. It is a square matrix of second-order partial derivatives of a scalar-valued function and is used primarily to analyze the local behavior around critical points.

For a two-variable function, the matrix is:\[H = \begin{bmatrix} f_{xx} & f_{xy} \ f_{xy} & f_{yy} \end{bmatrix}\]Calculating this matrix allows us to apply the Second Derivative Test through its determinant. Specifically, it examines the curvature of the surface that the function represents.
In the solution, the Hessian matrix provides valuable insight, indicating whether a critical point is a maximum, minimum, or a saddle point when evaluated at these points.

Saddle Point Concept

A saddle point is a type of critical point where the function does not have a local maximum or minimum. Instead, it has a behavior resembling a saddle, curving up in one direction and down in another. These points reveal that the region is neither purely top of a hill nor bottom of a valley.

When evaluating the point \((0, 0)\) for the function \( f(x, y) = e^{xy} \), the determinant of the Hessian is negative:

• This indicates that the point is a saddle point as per the Second Derivative Test.

A negative Hessian determinant confirms that the curvature of the graph changes direction around this point. Understanding saddle points is crucial for identifying areas of instability or change in multiple dimensions.