Critical Points Of F(x, Y) = X² + 2xy And Second Derivative Test

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In the realm of multivariable calculus, identifying critical points of a function is a fundamental step in understanding its behavior. These points, where the function's rate of change is zero or undefined, often correspond to local maxima, local minima, or saddle points. In this article, we will delve into the process of finding the critical points for the function f(x, y) = x² + 2xy, providing a comprehensive analysis and a step-by-step approach to solving this type of problem. Understanding critical points is crucial not only in mathematics but also in various fields such as physics, engineering, and economics, where optimization problems frequently arise.

The function we are examining, f(x, y) = x² + 2xy, is a quadratic function in two variables. Its graph is a surface in three-dimensional space, and the critical points correspond to the locations where the tangent plane to this surface is horizontal. To find these points, we need to employ the tools of partial derivatives and solve the system of equations that arise when setting these derivatives equal to zero. This process will reveal the coordinates (x, y) where the function's rate of change in both the x and y directions is zero. Let’s embark on this journey, and explore how critical points shape our understanding of functions and their applications.

Finding Critical Points Step-by-Step

1. Calculate the First Partial Derivatives

To begin, we need to compute the first partial derivatives of f(x, y) with respect to x and y. The partial derivative with respect to x, denoted as f_x, represents the rate of change of the function as x varies while y is held constant. Similarly, the partial derivative with respect to y, denoted as f_y, represents the rate of change as y varies while x is constant.

For our function, f(x, y) = x² + 2xy, we apply the rules of differentiation to find these partial derivatives:

  • f_x = ∂f/∂x = 2x + 2y
  • f_y = ∂f/∂y = 2x

These partial derivatives are crucial as they form the basis for identifying critical points. A critical point occurs where both f_x and f_y are simultaneously equal to zero, indicating a point where the function is neither increasing nor decreasing in either the x or y direction. These are the potential locations of local maxima, local minima, or saddle points.

2. Set the Partial Derivatives to Zero and Solve the System

Next, we set both partial derivatives equal to zero and solve the resulting system of equations:

  • f_x = 2x + 2y = 0
  • f_y = 2x = 0

From the second equation, f_y = 2x = 0, we immediately find that x = 0. This provides us with one coordinate of our critical point. To find the corresponding y-coordinate, we substitute x = 0 into the first equation:

  • 2(0) + 2y = 0
  • 2y = 0
  • y = 0

Thus, we have found that y = 0. Therefore, the solution to this system of equations is the point (0, 0). This point is our candidate for a critical point of the function f(x, y) = x² + 2xy.

3. Identify the Critical Point

Based on the solution of the system of equations, we have identified a single critical point for the function f(x, y) = x² + 2xy. This critical point is located at (0, 0). At this point, the function's rate of change in both the x and y directions is zero, suggesting that this could be a local maximum, a local minimum, or a saddle point. To determine the nature of this critical point, we need to further analyze the function's behavior around this point. This often involves calculating the second partial derivatives and applying the second derivative test, which we will explore in the subsequent sections.

The identification of the critical point (0, 0) is a significant step in understanding the function f(x, y) = x² + 2xy. It provides a specific location where the function's behavior may exhibit interesting characteristics. The next steps will involve delving deeper into the function's properties to classify the nature of this critical point and gain a more complete understanding of the function's overall behavior.

Calculate Second Derivatives

To classify the nature of the critical point (0, 0), we need to calculate the second partial derivatives of f(x, y) = x² + 2xy. These derivatives will help us understand the concavity of the function at the critical point, which is essential for determining whether it is a local maximum, local minimum, or saddle point. The second partial derivatives are:

  • f_xx = ∂²f/∂x²: The second partial derivative with respect to x, which measures the concavity of the function in the x-direction.
  • f_yy = ∂²f/∂y²: The second partial derivative with respect to y, which measures the concavity of the function in the y-direction.
  • f_xy = ∂²f/∂x∂y: The mixed partial derivative, which measures how the rate of change of f_x changes with respect to y, or equivalently, how the rate of change of f_y changes with respect to x.
  • f_yx = ∂²f/∂y∂x: The mixed partial derivative, which measures how the rate of change of f_y changes with respect to x, or equivalently, how the rate of change of f_x changes with respect to y.

For our function, we have already found the first partial derivatives:

  • f_x = 2x + 2y
  • f_y = 2x

Now, we differentiate these again to find the second partial derivatives:

  • f_xx = ∂(2x + 2y)/∂x = 2
  • f_yy = ∂(2x)/∂y = 0
  • f_xy = ∂(2x + 2y)/∂y = 2
  • f_yx = ∂(2x)/∂x = 2

Notice that f_xy and f_yx are equal, which is consistent with Clairaut's Theorem, which states that if the second partial derivatives are continuous, then the order of differentiation does not matter. These second partial derivatives provide crucial information for the second derivative test, which we will use to classify the critical point (0, 0).

Applying the Second Derivative Test

Now that we have calculated the second partial derivatives, we can apply the second derivative test to classify the critical point (0, 0). The second derivative test involves computing the discriminant, often denoted as D, which is defined as:

  • D = f_xx * f_yy - (f_xy)²

The value of D at the critical point, along with the value of f_xx, will determine the nature of the critical point. The rules for classification are as follows:

  1. If D > 0 and f_xx > 0, then the critical point is a local minimum.
  2. If D > 0 and f_xx < 0, then the critical point is a local maximum.
  3. If D < 0, then the critical point is a saddle point.
  4. If D = 0, the test is inconclusive, and other methods must be used to determine the nature of the critical point.

For our function, we have:

  • f_xx = 2
  • f_yy = 0
  • f_xy = 2

Now, we compute the discriminant D:

  • D = (2)(0) - (2)² = -4

Since D = -4 < 0, the second derivative test tells us that the critical point (0, 0) is a saddle point. This means that at (0, 0), the function has a minimum in one direction and a maximum in another direction. The surface of the function resembles a saddle near this point.

Conclusion

In this article, we have comprehensively analyzed the function f(x, y) = x² + 2xy to find and classify its critical points. We began by calculating the first partial derivatives and setting them equal to zero to find the critical point (0, 0). We then computed the second partial derivatives and applied the second derivative test. The discriminant D was found to be -4, which indicated that the critical point (0, 0) is a saddle point.

Understanding critical points and how to classify them is essential in multivariable calculus. It allows us to gain insights into the behavior of functions and their graphical representations. Saddle points, like the one we found, are particularly interesting as they represent points where the function exhibits different behaviors depending on the direction of approach. The process of finding critical points and classifying them has broad applications in various fields, including optimization problems in engineering, economics, and physics.

This analysis provides a thorough understanding of the critical points of the function f(x, y) = x² + 2xy. The step-by-step approach, from calculating partial derivatives to applying the second derivative test, is a valuable method for analyzing other multivariable functions and their critical points. By mastering these techniques, one can gain a deeper appreciation for the intricacies of calculus and its applications in the real world.