We are given the function $f(x, y) = x \sin y$ and asked to determine the nature of the critical point at $(0, 0)$.

Steps to determine the nature of the critical point:

1. Find the partial derivatives: To find the critical points and their nature, we start by computing the partial derivatives of $f(x, y)$ with respect to $x$ and $y$.

   • $f_x(x, y) = \frac{\partial}{\partial x} (x \sin y) = \sin y$
   • $f_y(x, y) = \frac{\partial}{\partial y} (x \sin y) = x \cos y$

2. Check the critical points: The critical points occur when both partial derivatives are zero:

   • $f_x(x, y) = 0$ implies $\sin y = 0$, so $y = n\pi$ where $n \in \mathbb{Z}$ (integer values).
   • $f_y(x, y) = 0$ implies $x \cos y = 0$. This gives two possibilities:
     • $x = 0$, or
     • $\cos y = 0$, meaning $y = (2n+1)\frac{\pi}{2}$.

   For the point $(0,0)$, we have $x = 0$ and $y = 0$, so this satisfies both conditions, confirming that $(0,0)$ is a critical point.

3. Second derivative test: We now use the second derivative test to classify the critical point. First, compute the second partial derivatives:

   • $f_{xx}(x, y) = 0$
   • $f_{yy}(x, y) = -x \sin y$
   • $f_{xy}(x, y) = \cos y$

   At $(0, 0)$, the second partial derivatives are:

   • $f_{xx}(0, 0) = 0$
   • $f_{yy}(0, 0) = 0$
   • $f_{xy}(0, 0) = 1$

   The Hessian determinant is given by: $D = f_{xx}f_{yy} - (f_{xy})^2 = (0)(0) - (1)^2 = -1$ Since $D < 0$, the critical point at $(0, 0)$ is a saddle point.

Conclusion:

The critical point $(0, 0)$ is a saddle point.

Would you like any further details or explanations?

Here are some related questions to deepen your understanding:

1. How do you calculate the Hessian matrix for a multivariable function?
2. What is the significance of the sign of the Hessian determinant?
3. Can a critical point with $D < 0$ ever be anything other than a saddle point?
4. How does the second derivative test differ from finding critical points in single-variable calculus?
5. What happens if the Hessian determinant is zero?

Tip: When using the second derivative test, always check the sign of the Hessian determinant to classify the critical point.