Question

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. $$ f(x, y)=\left(x^{2}+y^{2}\right)^{2 / 3} $$

Short answer

The critical point of function \(f(x, y) = (x^2 + y^2)^{2 / 3}\) is at the origin (0,0). However, the Second Partials Test or the Hessian determinant test fails to determine any relative extrema at this point as the determinant is 0.

Step-by-step solution

Compute the First Order Partial Derivatives

Calculate the partial derivatives with respect to x and y respectively. For this function they are given by \[f_x =\frac{4}{3}x(x^2+y^2)^{-1/3}\] and \[f_y = \frac{4}{3}y(x^2+y^2)^{-1/3}\]

Find the Critical Points

The critical points are where \(f_x = 0\) and \(f_y = 0\) or where these derivatives do not exist. Solving these two equations, you'll get that the critical point of f is (0,0).

Compute the Second Order Partial Derivatives

Calculate the second order partial derivatives. \[f_{xx}=\frac{4}{9}x(x^2+y^2)^{-4/3}\] \[f_{yy}=\frac{4}{9}y(x^2+y^2)^{-4/3}\] \[f_{xy}=f_{yx}=\frac{4}{9}y(x^2+y^2)^{-4/3}\]

Compute the Hessian Determinant

The determinant of Hessian matrix \(H(x, y)= D(x, y)=f_{xx}f_{yy}-(f_{xy})^2\). After inserting values \(D(0,0) = 0\).

Conduct the Second Partials Test

If D > 0 and \(f_{xx}(x_0, y_0) > 0\), f has a relative minimum at the point. If D > 0 and \(f_{xx}(x_0, y_0) < 0\), f has a relative maximum at the point. If D < 0, f has a saddle point at the point. If D = 0, the test is inconclusive. Here, the Hessian determinant at (0,0) is 0, so the test fails.

Key concepts

Understanding Partial Derivatives

In calculus, partial derivatives are used to measure how a function changes as each variable changes, while keeping all other variables constant. They are fundamental in the study of functions with more than one variable and are foundational in the fields of multivariable calculus and differential equations.

For a multivariable function such as \( f(x, y) \), the partial derivative with respect to x, often denoted as \( f_x \), signifies the rate at which the function's value changes as \( x \) changes, with \( y \) held fixed. Similarly, the partial derivative with respect to \( y \), denoted as \( f_y \), measures the change in \( f \) as \( y \) varies, with \( x \) being constant.
To find a function's critical points, we set its partial derivatives equal to zero because these points represent where the rate of change of the function ceases in the direction of either variable. For the cited exercise, the critical points are found where both \( f_x = 0 \) and \( f_y = 0 \), which leads to the discovery of a critical point at (0,0).

Applying the Second Partials Test

The Second Partials Test, also known as the Hessian criterion, is a method used to classify the critical points of a differentiable function. This test examines the second order partial derivatives of the function at a specific point.

Once you've located the critical points, it's essential to determine whether these points are relative maxima, minima, or saddle points. The Second Partials Test involves evaluating the second order partial derivatives,\( f_{xx} \), \( f_{yy} \), and the mixed partial, \( f_{xy} \), at the critical point. From these, one forms the Hessian determinant, denoted as \( D(x, y) \), and uses it to analyze the concavity of the function at the critical point.

A positive determinant \( (D > 0) \) may indicate a local minimum or maximum, depending on the sign of \( f_{xx} \). In contrast, a negative determinant \( (D < 0) \) signifies a saddle point. However, if the determinant is zero, as it is in this problem, the test fails to give conclusive results, and other methods must be used to classify the critical point.

Deciphering the Hessian Matrix

The Hessian matrix is a square matrix of all the second order partial derivatives of a scalar-valued function. In the context of the exercise, we denote it with \( H(x, y) \) for a function \( f(x, y) \). This matrix is a powerful tool used not only in calculus but also in optimization and differential geometry.

The Hessian matrix of a function describes the local curvature of that function in various directions. The determinant of the Hessian matrix, often used in the Second Partials Test as discussed earlier, provides information about the concavity and type of critical point by considering the second derivatives collectively. Therefore, establishing the Hessian determinant is a crucial step in analyzing a function's behavior at its critical points.

In this particular exercise, the Hessian matrix at the critical point (0,0) results in a determinant of zero, which unfortunately does not yield a definitive answer regarding the nature of the critical point through the Second Partials Test, indicating the need for additional investigation to fully classify it.