Given the function $f(x,y)=x^2-y^2$, which of the following statements correctly describes its local maxima, local minima, and saddle points?

• A. The function has a local maximum at $(0,0)$ and no saddle points.
• B. The function has a saddle point at $(0,0)$ and no local maxima or minima.
• C. The function has both a local maximum and a local minimum at $(0,0)$.
• D. The function has a local minimum at $(0,0)$ and no saddle points.

Answer: (B)

Verified step by step guidance

1. Step 1: To analyze the local maxima, minima, and saddle points of the function f(x, y) = x^2 - y^2, start by finding the critical points. Critical points occur where the partial derivatives of f with respect to x and y are both zero. Compute the partial derivatives: ∂f/∂x = 2x and ∂f/∂y = -2y.
2. Step 2: Set the partial derivatives equal to zero to find the critical points. Solve 2x = 0 and -2y = 0, which gives x = 0 and y = 0. Therefore, the critical point is at (0, 0).
3. Step 3: To classify the critical point, use the second derivative test. Compute the second partial derivatives: ∂²f/∂x² = 2, ∂²f/∂y² = -2, and the mixed partial derivative ∂²f/∂x∂y = 0.
4. Step 4: Calculate the determinant of the Hessian matrix, which is given by D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)². Substitute the values: D = (2)(-2) - (0)² = -4. Since D < 0, the critical point at (0, 0) is a saddle point.
5. Step 5: Conclude that the function f(x, y) = x^2 - y^2 has a saddle point at (0, 0) and no local maxima or minima. This matches the correct answer provided in the problem.