91Ó°ÊÓ

Partial derivatives are a cornerstone of multivariable calculus, providing a way to understand the rate of change of a function in relation to just one of its variables, while the other remains constant. This is similar to how derivatives function in single-variable calculus, but here, it's expanded to multiple dimensions.

For a function of two variables, like the one in our example, you will often calculate two partial derivatives:

• With respect to the first variable, say, \( x \), written as \( f_x \).
• With respect to the second variable, say, \( y \), written as \( f_y \).

To find these, you need to hold one variable constant and differentiate with respect to the other. In our example, for \( f(x, y) = e^x(x^2 - y^2) \), the first-order partial derivatives are:

• \( f_x = e^{x}(2x + x^2 - y^2) \), showing how \( f \) changes with \( x \).
• \( f_y = -2y e^{x} \), showing how \( f \) changes with \( y \).

Identifying critical points is a crucial step in analyzing a function to understand its local behavior – such as locating potential maxima, minima, or saddle points. Critical points in the context of multivariable functions are points where the first-order partial derivatives are zero.

These points are akin to the concept of turning points in single-variable calculus, marking where the function doesn’t increase or decrease immediately, indicating a potential maximum, minimum, or something in-between.

In our function example, we find that setting \( f_x = 0 \) results in \( 2x + x^2 - y^2 = 0 \). Simultaneously, setting \( f_y = 0 \) yields \( y = 0 \). Solving these equations, we find critical points at

• \( (0, 0) \)
• \( (-2, 0) \)

The second derivative test is an essential tool to distinguish the nature of critical points found in a function. By examining the second-order partial derivatives, this test helps determine if a critical point is a local maximum, minimum, or a saddle point.

To apply the second derivative test in multivariable calculus, you need:

• \( f_{xx} \) – the second partial derivative with respect to \( x \)
• \( f_{yy} \) – the second partial derivative with respect to \( y \)
• \( f_{xy} \) – the mixed partial derivative with respect to \( x \) and \( y \)

The determinant \( D \) is then calculated as \( D = f_{xx} f_{yy} - (f_{xy})^2 \).

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

A saddle point is a peculiar type of critical point that is neither a local maximum nor a local minimum. It is named because of its resemblance to a horse saddle, indicating regions that curve upwards and downwards.

A saddle point occurs when the test results in \( D < 0 \) in the second derivative test, reflecting a point where the function does not consistently increase or decrease in all directions. For instance, the point \( (0, 0) \) in the provided solution is a saddle point.

At this point, although it might seem the function is positioning itself at a critical position potentially leading to maximum or minimum values—by further evaluation, it's clear the directional rates of the function's change are mixed, creating a hyperbolic surface rather than a peak or valley. Understanding saddle points is crucial for optimizing complex systems, as they show direction-dependent behaviors.