91Ó°ÊÓ

Partial derivatives are crucial when dealing with multivariable calculus. They measure how a multivariable function changes as each individual variable changes, while all others are held constant. For the function given, \( f(x, y) = e^y - y e^x \), we calculate the first partial derivatives with respect to \( x \) and \( y \). This helps in finding where the function has stationary points, which could be potential local maxima, minima, or saddle points.

• The partial derivative with respect to \( x \), \( \frac{\partial f}{\partial x} = -ye^x \), captures how \( f \) changes with a small change in \( x \) while \( y \) is constant.
• The partial derivative with respect to \( y \), \( \frac{\partial f}{\partial y} = e^y - e^x \), tells us how \( f \) changes as \( y \) varies with \( x \) fixed.

To find critical points, we set these derivatives to zero, solving \( \frac{\partial f}{\partial x} = 0 \) and \( \frac{\partial f}{\partial y} = 0 \) at the same time.

This results in the critical point \( (0,0) \), meaning that at this point, the function "pauses" its change in both directions.

The Hessian matrix is a square matrix comprising all the second-order partial derivatives of a scalar-valued function. It is helpful in assessing the nature of critical points. For the function \( f(x, y) = e^y - ye^x \), the Hessian matrix is calculated to understand how these second derivatives behave together.

To build the Hessian matrix, we use:

• Second partial derivative with respect to \( x \), \( f_{xx} = \frac{\partial^2 f}{\partial x^2} \)
• Second partial derivative with respect to \( y \), \( f_{yy} = \frac{\partial^2 f}{\partial y^2} \)
• Mixed partial derivative, \( f_{xy} = \frac{\partial^2 f}{\partial x \partial y} \)

At the point \( (0, 0) \), we evaluated the Hessian matrix as: \[ \begin{bmatrix} 0 & -1 \ -1 & 1 \end{bmatrix} \] The value of these mixed second derivatives at our point aids in further classification using the determinant of the Hessian.

A saddle point, typically identified using the determinant of the Hessian matrix, is a critical point that is neither a local maximum nor a local minimum. In our analysis of the given function \( f(x, y) = e^y - ye^x \), we found that the point \( (0, 0) \) is a saddle point.

To comprehend this, consider the resulting Hessian determinant obtained as:
\[ D = f_{xx}f_{yy} - (f_{xy})^2 = (0)(1) - (-1)^2 = -1 \] Since \( D < 0 \), the surface of \( f(x, y) \) around the critical point \( (0, 0) \) looks like a saddle, curving upwards in one direction and downwards in another.

Thus, the point does not correspond to a local extremum but rather is atop a "ridge" or "saddle," demonstrating diverse curvatures.

The second derivative test in multivariable calculus aids in classifying the nature of critical points by examining the concavity of the function at these points. For a function of two variables, this involves the second partial derivatives arranged in the Hessian matrix.

Critical points, determined by first setting the first partial derivatives to zero, are then examined by this test. The determinant of the Hessian, \( D \), and the second partial derivatives give insight into the type of critical point:

• If \( D > 0 \) and \( f_{xx} > 0 \), the point is a local minimum.
• If \( D > 0 \) and \( f_{xx} < 0 \), the point is a local maximum.
• If \( D < 0 \), the critical point is a saddle point.

In the considered problem, we applied this test and found \( D = -1 \), leading to the conclusion that \( (0, 0) \) is a saddle point. This test is invaluable in identifying the nuances of the surface geometry at critical points.