Problem 26 - Calculus an Applied Approach

Question

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

Step-by-Step Solution

Answer

The only critical point found is (0,0). However, the Second Partials Test fails at this point, which means the test cannot determine whether this point is a relative maximum, relative minimum, or saddle point.

Step 1: Compute the partial derivatives of the function

To find the critical points, first find the first partial derivatives of \(f\). Given \(f(x, y)=\sqrt{x^{2}+y^{2}}\), the partial derivatives are computed as follows:\n\n\[f_x = \frac{{x}}{{\sqrt{x^2 + y^2}}} \]\[f_y = \frac{{y}}{{\sqrt{x^2 + y^2}}}\]

Step 2: Find the critical points

The critical points occur when both partial derivatives are equal to zero or undefined, thus set \(f_x = 0\) and \(f_y = 0\) and solve for \(x\) and \(y\).\n However, these derivatives don't have any real solutions. In each of \(f_x\) and \(f_y\), we can see that it's impossible for either x or y alone to cancel out and yet both exist in the denominator. This means the derivative does not exist at the origin (0,0) - thus (0,0) is a critical point. So, the only critical point for this function is (0,0).

Step 3: Second Partials Test

Next, calculate the second order partial derivatives:\[f_{xx} = \frac{{y^2}}{{\left( x^2 + y^2 \right)^{3/2}}}\]\[f_{yy} = \frac{{x^2}}{{\left( x^2 + y^2 \right)^{3/2}}}\]\[f_{xy} = f_{yx} = -\frac{{xy}}{{\left( x^2 + y^2 \right)^{3/2}}}\]The determinant of the Hessian matrix (D) is computed as follows:\[D = f_{xx} f_{yy} - \left( f_{xy} \right)^2 = 0\]at the origin (0,0). So, the second partials test fails to determine the nature of the critical point (0,0).

Key Concepts

Partial Derivatives

When dealing with functions of several variables, such as functions in calculus with variables \(x\) and \(y\), partial derivatives are used to understand how the function changes with respect to one variable at a time. In this exercise, we consider the function \(f(x, y)=\sqrt{x^{2}+y^{2}}\). The partial derivative with respect to \(x\) is calculated by differentiating \(f(x, y)\) while treating \(y\) as a constant, resulting in \(f_x = \frac{{x}}{{\sqrt{x^2 + y^2}}} \). Similarly, the partial derivative with respect to \(y\), denoted as \(f_y\), is found by differentiating with \(x\) considered a constant, giving us \(f_y = \frac{{y}}{{\sqrt{x^2 + y^2}}} \).
Partial derivatives are powerful tools for checking changes along specific directions for multivariable functions, a key step in finding critical points.

Critical Points

Critical points are points in the domain of a function where the gradient (or all first partial derivatives) is zero or undefined. These points are vital in calculus because they are potential locations for local minima, maxima, or saddle points. In our given function \(f(x, y)=\sqrt{x^{2}+y^{2}}\), we determine the critical points by setting \(f_x = 0\) and \(f_y = 0\). However, as noted in the solution, these equations do not yield any real solution except at the origin due to undefined outcomes. Thus, the function only has a single critical point at \((0,0)\).
Finding critical points is crucial, as it helps in identifying where a function might change its behavior.

Second Partials Test

The Second Partials Test is applied to determine the nature of a critical point once we have located it. This test assesses whether a critical point is a local minimum, maximum, or saddle point.

• The function's second partial derivatives, along with a specific determinant formula, are involved in this test.
• The determinant, referred to as the Hessian determinant \(D\), is defined as \(D = f_{xx} f_{yy} - (f_{xy})^2\).

For the critical point at the origin \((0,0)\), calculations show \(D = 0\). Within the scope of the Second Partials Test, a zero determinant indicates the test is inconclusive. Hence, at \((0,0)\), we cannot use this test to determine the function's extremum nature. Understanding how to apply this test and its limitations is essential for identifying the behavior of critical points.

Hessian Matrix

The Hessian Matrix is a square matrix that contains all the second-order partial derivatives of a function. It is used extensively in optimization and is pivotal in the Second Partials Test.

• For a function \(f(x, y)\), the Hessian matrix is structured as follows: \[ H = \begin{bmatrix} f_{xx} & f_{xy} \f_{yx} & f_{yy} \end{bmatrix} \]
• In our exercise, \(f_{xx} = \frac{{y^2}}{{(x^2 + y^2)^{3/2}}}\), \(f_{yy} = \frac{{x^2}}{{(x^2 + y^2)^{3/2}}}\), and \(f_{xy} = f_{yx} = -\frac{{xy}}{{(x^2 + y^2)^{3/2}}}\).

The determinant of this Hessian matrix is vital for assessing the second order conditions in a multivariable function. Despite its power, if at a critical point the determinant equals zero, like our case at \((0,0)\), the Second Partials Test does not provide information about the function's behavior at that point. Learning how to compute and interpret the Hessian matrix enhances the understanding of multivariable calculus.