Local Extrema Of F(x, Y) = X³ + Y³ - 5x - 7y² + 9 A Comprehensive Analysis

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Introduction

In this comprehensive exploration, we will delve into the mathematical function f(x, y) = x³ + y³ - 5x - 7y² + 9. Our primary objective is to meticulously determine and classify the local extrema of this function. Local extrema, encompassing local maxima and minima, represent critical points where the function's value is either at its highest (maximum) or lowest (minimum) within a specific neighborhood. To achieve this, we will systematically employ the principles of multivariable calculus, utilizing partial derivatives and the Hessian matrix to identify and characterize these crucial points.

Understanding Local Extrema

Before embarking on the detailed calculations, it is imperative to establish a clear understanding of what local extrema signify within the context of a multivariable function. A local maximum occurs at a point (x₀, y₀) if the function value f(x₀, y₀) is greater than or equal to the function value at all points in a small neighborhood around (x₀, y₀). Conversely, a local minimum occurs at a point (x₀, y₀) if the function value f(x₀, y₀) is less than or equal to the function value at all points in a small neighborhood around (x₀, y₀). These points are critical in understanding the behavior and characteristics of the function's surface.

Calculating Partial Derivatives

To pinpoint the potential locations of local extrema, we must first compute the partial derivatives of the function f(x, y) with respect to both x and y. These partial derivatives, denoted as ∂f/∂x and ∂f/∂y, respectively, provide valuable insights into the function's rate of change along the x and y axes. The partial derivative ∂f/∂x is obtained by differentiating f(x, y) with respect to x while treating y as a constant. Similarly, ∂f/∂y is obtained by differentiating f(x, y) with respect to y while treating x as a constant. For the given function f(x, y) = x³ + y³ - 5x - 7y² + 9, the partial derivatives are calculated as follows:

  • ∂f/∂x = 3x² - 5
  • ∂f/∂y = 3y² - 14y

Identifying Critical Points

Critical points are the cornerstone of identifying local extrema. These are the points (x, y) where both partial derivatives simultaneously equal zero, or where one or both partial derivatives are undefined. In essence, critical points represent locations where the function's slope is zero in both the x and y directions, making them potential candidates for local maxima or minima. To find the critical points of our function, we set both partial derivatives equal to zero and solve the resulting system of equations:

  • 3x² - 5 = 0
  • 3y² - 14y = 0

Solving the first equation, 3x² - 5 = 0, for x yields:

  • x² = 5/3
  • x = ±√(5/3)

Solving the second equation, 3y² - 14y = 0, for y yields:

  • y(3y - 14) = 0
  • y = 0 or y = 14/3

Thus, we have identified four critical points: (√(5/3), 0), (-√(5/3), 0), (√(5/3), 14/3), and (-√(5/3), 14/3).

The Hessian Matrix and the Second Derivative Test

Having identified the critical points, the next crucial step is to classify these points as either local maxima, local minima, or saddle points. To achieve this classification, we employ the Hessian matrix and the second derivative test. The Hessian matrix is a square matrix of second-order partial derivatives of a multivariable function. For our function f(x, y), the Hessian matrix, denoted as H(x, y), is defined as:

H(x, y) = | ∂²f/∂x²   ∂²f/∂x∂y |
          | ∂²f/∂y∂x   ∂²f/∂y² |

We need to compute the second-order partial derivatives:

  • ∂²f/∂x² = 6x
  • ∂²f/∂y² = 6y - 14
  • ∂²f/∂x∂y = ∂²f/∂y∂x = 0

Thus, the Hessian matrix for our function is:

H(x, y) = | 6x       0      |
          | 0       6y - 14 |

The second derivative test involves evaluating the determinant of the Hessian matrix, denoted as D(x, y), and the second-order partial derivative with respect to x, ∂²f/∂x², at each critical point. The determinant of the Hessian matrix is calculated as:

  • D(x, y) = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²
  • D(x, y) = (6x)(6y - 14) - 0²
  • D(x, y) = 36x(y - 14/6) = 36x(y - 7/3)

The classification of critical points is based on the following criteria:

  1. If D(x, y) > 0 and ∂²f/∂x² > 0, then the critical point is a local minimum.
  2. If D(x, y) > 0 and ∂²f/∂x² < 0, then the critical point is a local maximum.
  3. If D(x, y) < 0, then the critical point is a saddle point.
  4. If D(x, y) = 0, the test is inconclusive.

Classifying the Critical Points

Now, we will apply the second derivative test to classify each of the four critical points we identified earlier.

1. Critical Point: (√(5/3), 0)

  • D(√(5/3), 0) = 36√(5/3)(0 - 7/3) = -84√(5/3) < 0

Since D(√(5/3), 0) < 0, the critical point (√(5/3), 0) is a saddle point. Saddle points represent points where the function has a minimum in one direction and a maximum in another direction.

2. Critical Point: (-√(5/3), 0)

  • D(-√(5/3), 0) = 36(-√(5/3))(0 - 7/3) = 84√(5/3) > 0
  • ∂²f/∂x²(-√(5/3), 0) = 6(-√(5/3)) = -6√(5/3) < 0

Since D(-√(5/3), 0) > 0 and ∂²f/∂x²(-√(5/3), 0) < 0, the critical point (-√(5/3), 0) is a local maximum. This point represents a location where the function attains its highest value within a specific neighborhood.

3. Critical Point: (√(5/3), 14/3)

  • D(√(5/3), 14/3) = 36√(5/3)(14/3 - 7/3) = 36√(5/3)(7/3) = 84√(5/3) > 0
  • ∂²f/∂x²(√(5/3), 14/3) = 6√(5/3) > 0

Since D(√(5/3), 14/3) > 0 and ∂²f/∂x²(√(5/3), 14/3) > 0, the critical point (√(5/3), 14/3) is a local minimum. This point represents a location where the function attains its lowest value within a specific neighborhood.

4. Critical Point: (-√(5/3), 14/3)

  • D(-√(5/3), 14/3) = 36(-√(5/3))(14/3 - 7/3) = 36(-√(5/3))(7/3) = -84√(5/3) < 0

Since D(-√(5/3), 14/3) < 0, the critical point (-√(5/3), 14/3) is a saddle point.

Conclusion

In conclusion, our comprehensive analysis of the function f(x, y) = x³ + y³ - 5x - 7y² + 9 has revealed the following local extrema:

  • One local maximum: Located at the point (-√(5/3), 0).
  • One local minimum: Located at the point (√(5/3), 14/3).
  • Two saddle points: Located at the points (√(5/3), 0) and (-√(5/3), 14/3).

This detailed determination of local extrema provides a thorough understanding of the function's behavior and characteristics. The application of partial derivatives, the Hessian matrix, and the second derivative test has proven instrumental in accurately identifying and classifying these critical points. This methodology is widely applicable in various fields, including optimization problems, physics, and engineering, where understanding the extrema of functions is crucial for effective decision-making and problem-solving.

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