Problem 2 - APEX Calculus

Question

T/F: A point \(P\) is a critical point of \(f\) if \(f_{x}\) and \(f_{y}\) are both 0 at \(P\).

Step-by-Step Solution

Answer

True.

Step 1: Define Critical Point

A critical point of a function of two variables, \( f(x, y) \), occurs where both of its partial derivatives, \( f_x \) (with respect to \( x \)) and \( f_y \) (with respect to \( y \)), are zero, or where at least one of the partial derivatives is undefined.

Step 2: Analyze Given Condition

In the exercise, the statement claims that \( P \) is a critical point if \( f_x = 0 \) and \( f_y = 0 \) at \( P \). This condition matches the definition of a critical point as both partial derivatives being zero at a point satisfies one of the conditions for a critical point.

Step 3: Conclusion

Since the condition given (both partial derivatives being zero) directly aligns with one criterion for a critical point, the statement is true.

Key Concepts

Partial Derivatives

Partial derivatives are a type of derivative used in the context of functions of more than one variable. Consider a function \( f(x, y) \), which depends on two variables \( x \) and \( y \). A partial derivative measures how the function changes as one of these variables changes, while the other variable is kept constant. This is a crucial concept in multivariable calculus because it helps us analyze the behavior of complex functions.

• The partial derivative of \( f \) with respect to \( x \), denoted as \( f_x \), tells us how \( f \) changes as \( x \) changes while \( y \) remains constant.
• Similarly, the partial derivative of \( f \) with respect to \( y \), denoted as \( f_y \), shows how \( f \) changes as \( y \) changes while \( x \) remains constant.

To find these derivatives, we treat one variable as a constant and differentiate with respect to the other. Understanding partial derivatives is essential for locating critical points, optimizing functions, and more.

Functions of Two Variables

Functions of two variables occur frequently in real-world applications, where outcomes depend on more than one parameter, like temperature and pressure for a gas law. Such functions are of the form \( f(x, y) \), where \( x \) and \( y \) are the inputs to the function. The output, \( f(x, y) \), could be anything from an elevation in a landscape to profit in a business scenario.

These functions can be visualized as surfaces in three-dimensional space. As a result:

• The graph of \( f(x, y) \) can be continuous, defined over a domain in the \( xy \)-plane.
• The shape and "curvature" of the surface tells us how \( f \) behaves, which can be analyzed using partial derivatives.

Working with these functions requires understanding how changes in input variables affect the overall system. This can mean parameter variations in mathematical models or real staging in engineering or natural processes.

Conditions for Critical Points

Identifying critical points in functions of two variables is a critical step in optimizing or understanding the behavior of these functions. A critical point occurs where the function levels out, meaning there's no immediate increase or decrease in any direction. These are places where potential maxima, minima, or saddle points exist.

To find critical points, we use the condition that the partial derivatives of the function must both be zero:

• If both \( f_x = 0 \) and \( f_y = 0 \), the point is a candidate for a critical point.
• Additionally, if either of the partial derivatives does not exist at a point, it may also be a critical point.

Assessing whether these points are indeed minima or maxima, or simply flat points, requires further analysis, sometimes involving the second derivatives or the Hessian matrix. However, the initial step always starts with checking where the partial derivatives vanish.