Problem 24 - College Algebra and 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 \(\left(x_{0}, y_{0}\right)\). $$ f_{x x}\left(x_{0}, y_{0}\right)=25, f_{v v}\left(x_{0}, y_{0}\right)=8, f_{x v}\left(x_{0}, y_{0}\right)=10 $$

Step-by-Step Solution

Answer

There is a relative minimum at the critical point \( x_{0}, y_{0} \) of the function \( f(x, y) \).

Step 1: Identify given variables

Here, the second order partial derivatives at point \( x_{0}, y_{0} \) are given. They are: \( f_{xx}(x_{0},y_{0}) = 25, f_{yy}(x_{0},y_{0}) = 8 \) and \( f_{xy}(x_{0},y_{0}) = 10 \)

Step 2: Compute formula for D

Now, using these values, compute D: \( D = f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0}) - [f_{xy}(x_{0},y_{0})]^{2} = 25 * 8 - 10^{2} = 200 - 100 = 100 \)

Step 3: Determine nature of critical point

Since D > 0 and \( f_{xx}(x_{0},y_{0}) > 0 \), there is a relative minimum at the point \( x_{0}, y_{0} \).

Key Concepts

Partial Derivatives

Partial derivatives are essential when analyzing functions of multiple variables. They help us understand how these functions behave in different directions. For a function of two variables, like our exercise's function \(f(x, y)\), partial derivatives allow us to examine changes in the function along the \(x\) and \(y\) axes separately.

To find a partial derivative with respect to \(x\), we denote it as \(f_{x}\) or sometimes \(\frac{\partial f}{\partial x}\). It involves differentiating the function while treating \(y\) as a constant. This process shows how the function changes as \(x\) shifts, keeping \(y\) steady. Similarly, the partial derivative with respect to \(y\) is denoted as \(f_{y}\) or \(\frac{\partial f}{\partial y}\), and it examines changes in \(f\) as \(y\) changes.

Understanding partial derivatives is crucial for identifying critical points — locations where the function's rate of change is zero. These points can be potential candidates for relative minimums, maximums, or saddle points. By mastering partial derivatives, students can effectively navigate and analyze multi-variable functions.

Second Order Derivatives

Second order derivatives dive deeper into the function's behavior, offering more insight than first order derivatives. For a function like \(f(x, y)\), they involve taking partial derivatives of the partial derivatives. Specifically, \(f_{xx}\) is the second partial derivative with respect to \(x\), \(f_{yy}\) is with respect to \(y\), and \(f_{xy}\) or \(f_{yx}\) are mixed partial derivatives.

The second order derivatives are essential to determine how the function curves or bends at critical points. They help construct the Hessian matrix, which serves to compute the discriminant \(D\). This discriminant is calculated as:

\[ D = f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0}) - [f_{xy}(x_{0},y_{0})]^2 \]

Such a calculation reveals the nature of the critical point. This approach is key in higher mathematics and physics to analyze the stability and configuration of systems.

Relative Minimum

A relative minimum is one of the outcomes when analyzing critical points of a function. It is a point where the function reaches a low value compared to its nearby points. Identifying a relative minimum involves several criteria:

• The discriminant \(D\) must be greater than zero \((D > 0)\).
• The second partial derivative with respect to the \(x\) (\(f_{xx}\)) must be greater than zero \((f_{xx} > 0)\).

When both these conditions are satisfied, the point in question is verified as a relative minimum.

This solution to the original exercise verifies the presence of a relative minimum at the critical point \((x_{0}, y_{0})\), as calculations show \(D = 100\) (which is positive), and \(f_{xx} = 25\) (again, positive). Understanding how these criteria work is essential for anyone studying multivariable calculus, as it aids in assessing shapes and characteristics of graphs.