Problem 23 - Calculus an Applied Approach

Question

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right) .\) $$ f_{x x}\left(x_{0}, y_{0}\right)=-9, f_{y y}\left(x_{0}, y_{0}\right)=6, f_{x y}\left(x_{0}, y_{0}\right)=10 $$

Step-by-Step Solution

Answer

The function \(f(x, y)\) has a saddle point at \(\left(x_{0}, y_{0}\right).\

Step 1: Compute Hessian Matrix

In order to classify a critical point, we first have to compute the Hessian matrix, which is a square matrix of second-order mixed partial derivatives of our function \(f(x, y)\). The Hessian matrix for \(f(x, y)\) is given by: \[H(x, y)=\left[\begin{array}{cc}f_{x x} & f_{x y} \ f_{y x} & f_{y y}\end{array}\right]\] Hence, the Hessian Matrix at the point \(\left(x_{0}, y_{0}\right)\) would be: \[H(x_0, y_0)=\left[\begin{array}{cc}-9 & 10 \ 10 & 6\end{array}\right]\] Since partial derivative is commutative, \(f_{x y} = f_{y x}\), we can interchange the off-diagonal elements.

Step 2: Find the determinant of the Hessian matrix

The determinant of the Hessian matrix is: \(D = f_{x x} f_{y y} - (f_{x y})^2\). Substituting given values, we get: \[D = \left(-9\right)*6 - (10)^2 = -54 - 100 = -154\]

Step 3: Classify the critical point

Now, we classify the critical point \(\left(x_{0}, y_{0}\right)\). If \(D > 0\) and \(f_{x x} < 0\), the function has a relative maximum at this point. If \(D > 0\) and \(f_{x x} > 0\), the function has a relative minimum at this point. If \(D < 0\), then the function has a saddle point at this point. Since we got \(D = -154\), and since \(f_{x x} < 0\), it is a saddle point.

Key Concepts

Understanding Saddle Points

A saddle point is an interesting concept in multivariable calculus. When dealing with functions of two variables, a saddle point occurs where the surface of the function resembles a saddle. This means that at a saddle point, the function does not possess a maximum or minimum, but rather a point of inflection. This inflection leads to the function appearing to rise in one direction and descend in another from the saddle point. In terms of the Hessian matrix, if the determinant is less than zero, this indicates a saddle point. This is because the mixed signs in the derivative behavior create the unique saddle shape.

Thus, to identify a saddle point, one crucial step is calculating the determinant of the Hessian matrix and checking if it is negative. This is precisely what happened in the example provided, where the determinant was found to be \(-154\). This confirms the presence of a saddle point at the critical point \((x_0, y_0)\).

Identifying Critical Points

Critical points in a multivariable function are points where the gradient — a vector of partial derivatives — is zero or undefined. These points are crucial because they help in determining where the function could have a minimum, maximum, or saddle point.

To find critical points, one usually sets the first-order partial derivatives equal to zero and solves for the variables. These solutions provide the critical points of the function. Once these points are located, they are further analyzed using the second-order partial derivatives and other criteria.

Hemming in on these key areas allows further classification of the point, such as determining if it's merely a saddle point, a relative maximum, or a relative minimum, as seen in the example exercise. Here, the obtained critical point necessitated further study using the Hessian matrix to finally identify it as a saddle point.

The Role of Second-Order Partial Derivatives

Second-order partial derivatives are central in analyzing the behavior of functions at critical points. Typically analyzed through the Hessian matrix, these derivatives offer insights into the concavity and interactions between variables in the function.

The individual components of the Hessian — such as \(f_{xx}, f_{yy},\) and \(f_{xy} = f_{yx}\) — convey key information. The diagonal entries, \(f_{xx}\) and \(f_{yy}\), provide insight into how the function curves in each direction, while the off-diagonal entries, \(f_{xy}\) or \(f_{yx}\), show how the variables impact each other.

Particularly, testing these with the determinant helps in classifying the nature of critical points, such as revealing if a point is maximum, minimum, or a saddle. The exercise solution highlighted the calculation of these derivatives and their role in determining the Hessian determinant, ultimately leading to the identification of a saddle point.