Question

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

Short answer

The function \(f(x, y) = \sqrt{x^{2}+y^{2}}\) has a critical point at (0,0), but the Second-Partials Test fails at this point because the second partial derivatives are undefined.

Step-by-step solution

Compute the first derivatives

First, compute the first partial derivatives of the function \(f(x, y)\) with respect to \(x\) and \(y\). The partial derivative with respect to \(x\) is given by \(f_{x}(x,y) = x/ \sqrt{x^{2}+y^{2}}\). The partial derivative with respect to \(y\) is \(f_{y}(x,y) = y/ \sqrt{x^{2}+y^{2}}\).

Find the critical points

The critical points are where the first partial derivatives are equal to zero or undefined. For \(f_x(x, y)\) and \(f_y(x, y)\), both are undefined at (0,0), which is the only critical point of this function.

Compute the second derivatives

Next, compute the second partial derivatives of the function. The second partial derivative with respect to \(x\) is given by \(f_{xx}(x,y) = y^{2}/(x^{2}+y^{2})^{3/2}\), and the one with respect to \(y\) gives \(f_{yy}(x,y)= x^{2}/(x^{2}+y^{2})^{3/2}\). The mixed second partial derivatives are \(f_{xy}(x,y)= f_{yx}(x,y)= -xy/(x^{2}+y^{2})^{3/2}\).

Apply the Second-Partials Test

The Second-Partials Test requires computing the determinant of the Hessian matrix, \(D(x, y) = f_{xx}f_{yy} - (f_{xy})^2\). At the critical point (0,0), all second partial derivatives are undefined, and thus, the Second-Partials test cannot be applied.