Find and classify the critical points of the function f(x, y) = x sin y.
A) local maxima at (0, n $\pi$) for even integers n, local minimum at (0, n $\pi$) for odd integers n
B) local minima at (0, n $\pi$) for even integers n, local maximum at (0, n $\pi$) for odd integers n
C) saddle points at all points (0, n $\pi$) for integers n
D) only critical point is (0, 0), a saddle point
E) local minima at (0, n $\pi$) for nonzero integers n, saddle point at (0, 0)

Question

Quizplus · Accepted Answer

To find the critical points of $f(x, y) = x \sin y$, we first find the partial derivatives: $f_x = \sin y$ and $f_y = x \cos y$. Setting these equal to zero, we find that $f_x = 0$ for all $y$, and $f_y = 0$ when $x = 0$ or $\cos y = 0$. However, $f_x = 0$ does not depend on $x$, so we focus on $f_y = 0$, which gives critical points at $x = 0$ and $y = n\pi$ for integers $n$. To classify these points, we examine the second derivatives and the Hessian matrix. The second derivatives are $f_{xx} = 0$, $f_{yy} = -x \sin y$, and $f_{xy} = \cos y$. At $x = 0$, $f_{yy} = 0$, making the Hessian determinant $0$, which does not conclusively classify these points. However, considering the nature of the function and its symmetry, we can infer that these points do not lead to local maxima or minima but rather to changes in concavity, suggesting they are saddle points. Thus, the correct classification is that all points $(0, n\pi)$ for integers $n$ are saddle points.

Find and Classify the Critical Points of the Function F(x π\piπ

Find and Classify the Critical Points of the Function F(x $\pi$