Problem 22 - 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 \(\left(x_{0}, y_{0}\right) .\) $$ f_{x x}\left(x_{0}, y_{0}\right)=-3, f_{y y}\left(x_{0}, y_{0}\right)=-8, f_{x y}\left(x_{0}, y_{0}\right)=2 $$

Step-by-Step Solution

Answer

The critical point \(\left(x_{0}, y_{0}\right)\) is a relative maximum of the function.

Step 1: Setting up the Hessian determinant

The Hessian determinant \(D\) is defined as \(D = f_{xx}f_{yy} - (f_{xy})^2\). Here, it's necessary to just plug in the given values of the second partial derivatives into this formula.

Step 2: Calculating the Hessian determinant

Substituting the given values into the formula, it gives \(D = (-3)\cdot(-8) - (2)^2 = 24 - 4 = 20\).

Step 3: Analyzing the result

The determinant \(D\) is positive, and also both \(f_{xx}=-3\) and \(f_{yy}=-8\) are negative, which implies that the given point \(\left(x_{0}, y_{0}\right)\) is a relative maximum of the function \(f(x, y)\).

Key Concepts

Hessian Determinant

The Hessian Determinant is a powerful tool used in calculus to determine the nature of critical points of a two-variable function. It provides insight into whether a critical point is a relative maximum, a relative minimum, or a saddle point. This determinant is calculated using second partial derivatives and is represented by the formula: \[ D = f_{xx}f_{yy} - (f_{xy})^2 \] Here, \(f_{xx}\), \(f_{yy}\), and \(f_{xy}\) denote the second partial derivatives of the function \(f(x, y)\).

• The first term, \(f_{xx}f_{yy}\), involves multiplying the second derivative with respect to \(x\) by the second derivative with respect to \(y\).
• The second term, \((f_{xy})^2\), is the square of the mixed derivative.

Understanding the signs and values of these components helps in characterizing the critical point. A positive Hessian determinant might indicate a relative extremum or insufficient information depending on the signs of \(f_{xx}\) and \(f_{yy}\).

Second Partial Derivatives

Second Partial Derivatives offer deeper insight into the behavior of functions regarding changes in multiple variables.

• The derivative \(f_{xx}\) refers to the change in \(f_x\) (the derivative with respect to \(x\)) as \(x\) itself changes.
• Similarly, \(f_{yy}\) describes the change in \(f_y\) when \(y\) shifts.
• Lastly, \(f_{xy}\) (or \(f_{yx}\)) measures how the slope with respect to \(x\) alters as \(y\) shifts.

These derivatives are crucial for calculating the Hessian determinant. They can provide valuable insights about the curvature and the direction of concavity or convexity of the function at a certain critical point. By understanding these, one can interpret the shape and behavior of the graph in the vicinity of critical points.

Relative Maximum

A Relative Maximum occurs at a critical point where the function attains the highest value in a neighborhood. For a function of two variables, a critical point is classified as a relative maximum under specific conditions that involve the second partial derivatives and the calculated Hessian determinant.

• The Hessian determinant \(D\) must be greater than zero.
• Both \(f_{xx}\) and \(f_{yy}\) need to be negative, indicating that the function is curving downward in both directions from the point.

In our original exercise, both these criteria are met since \(D = 20\), \(f_{xx} = -3\), and \(f_{yy} = -8\). Therefore, the critical point is identified as a relative maximum. This tells us that in the very immediate neighborhood around this point, the function reaches a peak.

Saddle Point

A Saddle Point is an interesting type of critical point where the function achieves neither a relative maximum nor a minimum. Instead, the surface in the vicinity of this point curves upwards in one direction and downwards in another, resembling a horse’s saddle. To classify a saddle point using the Hessian determinant and second partial derivatives:

• The Hessian determinant \(D\) must be less than zero.

This change in sign is key to recognizing a saddle point. It tells us that the surface's curvature differs in two perpendicular directions at the point. Although the original exercise does not exemplify a saddle point, understanding this classification is important for distinguishing various behaviors at critical points.