23. Let f = f(x, y) be a twice continuously differentiable function. Let ∂ ∂xf(a, b) = 0, ∂ ∂yf(a, b) = 0, where a neighborhood of the point (a, b) is in the domain of the function f. Suppose that f attains a local maximum at (a, b). Which of the following statements is true? (A) ∂ 2 ∂x2 f(a, b) < 0, ∂ 2 ∂y2 f(a, b) < 0, ∂ 2 ∂x2 f(a, b) ∂ 2 ∂y2 f(a, b) −  ∂ 2 ∂x∂yf(a, b) 2 > 0 (B) ∂ 2 ∂x2 f(a, b) > 0, ∂ 2 ∂y2 f(a, b) > 0, ∂ 2 ∂x2 f(a, b) ∂ 2 ∂y2 f(a, b) −  ∂ 2 ∂x∂yf(a, b) 2 > 0 (C) ∂ 2 ∂x2 f(a, b) > 0, ∂ 2 ∂y2 f(a, b) > 0, ∂ 2 ∂x2 f(a, b) ∂ 2 ∂y2 f(a, b) −  ∂ 2 ∂x∂yf(a, b) 2 < 0 (D) ∂ 2 ∂x2 f(a, b) < 0, ∂ 2 ∂y2 f(a, b) < 0, ∂ 2 ∂x2 f(a, b) ∂ 2 ∂y2 f(a, b) −  ∂ 2 ∂x∂yf(a, b) 2 < 0

Given that \( f(x, y) \) is a twice continuously differentiable function and \( f \) attains a local maximum at the point \( (a, b) \), where the first partial derivatives \( \frac{\partial f}{\partial x}(a, b) = 0 \) and \( \frac{\partial f}{\partial y}(a, b) = 0 \), we are interested in the behavior of the second partial derivatives to determine which of the provided statements is true.

### Step 1: Analyze the second derivative test for functions of two variables

For a function \( f(x, y) \) of two variables, the second derivative test involves the Hessian matrix \( H \) at the critical point \( (a, b) \):

\[
H = \begin{pmatrix}
\frac{\partial^2 f}{\partial x^2}(a, b) & \frac{\partial^2 f}{\partial x \partial y}(a, b) \\
\frac{\partial^2 f}{\partial y \partial x}(a, b) & \frac{\partial^2 f}{\partial y^2}(a, b)
\end{pmatrix}
\]

The discriminant \( D \) is given by:

\[
D = \frac{\partial^2 f}{\partial x^2}(a, b) \cdot \frac{\partial^2 f}{\partial y^2}(a, b) - \left(\frac{\partial^2 f}{\partial x \partial y}(a, b)\right)^2
\]

### Step 2: Applying the second derivative test

- **Case 1: Local Maximum**
  - If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2}(a, b) < 0 \) and \( \frac{\partial^2 f}{\partial y^2}(a, b) < 0 \), then \( f \) has a local maximum at \( (a, b) \).

- **Case 2: Local Minimum**
  - If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2}(a, b) > 0 \) and \( \frac{\partial^2 f}{\partial y^2}(a, b) > 0 \), then \( f \) has a local minimum at \( (a, b) \).

- **Case 3: Saddle Point**
  - If \( D < 0 \), then \( (a, b) \) is a saddle point, regardless of the signs of \( \frac{\partial^2 f}{\partial x^2}(a, b) \) and \( \frac{\partial^2 f}{\partial y^2}(a, b) \).

- **Case 4: Inconclusive**
  - If \( D = 0 \), the test is inconclusive.

### Step 3: Identifying the correct statement

Given that \( f \) attains a local maximum at \( (a, b) \), we must have:

- \( \frac{\partial^2 f}{\partial x^2}(a, b) < 0 \)
- \( \frac{\partial^2 f}{\partial y^2}(a, b) < 0 \)
- \( D > 0 \), which means:
\[
\frac{\partial^2 f}{\partial x^2}(a, b) \cdot \frac{\partial^2 f}{\partial y^2}(a, b) > \left(\frac{\partial^2 f}{\partial x \partial y}(a, b)\right)^2
\]

This corresponds to **option (A)**:

\[
(A) \,\, \frac{\partial^2 f}{\partial x^2}(a, b) < 0, \, \frac{\partial^2 f}{\partial y^2}(a, b) < 0, \, \frac{\partial^2 f}{\partial x^2}(a, b) \cdot \frac{\partial^2 f}{\partial y^2}(a, b) - \left(\frac{\partial^2 f}{\partial x \partial y}(a, b)\right)^2 > 0
\]

**Answer**: The correct statement is **(A)**.

---

Would you like to explore any additional details on this topic? Here are some related questions:

1. How does the Hessian matrix help in determining the nature of critical points?
2. Can you explain why \( D > 0 \) indicates a local extremum?
3. What are the implications if \( D = 0 \)?
4. How does the second derivative test differ in one-dimensional and multi-dimensional contexts?
5. What is the significance of the mixed partial derivative in the Hessian matrix?
6. Can you give an example where \( D < 0 \) and explain why it results in a saddle point?
7. How does this analysis change for functions of three variables?
8. What are some practical applications of the second derivative test in optimization problems?

**Tip**: Always verify the sign of the second derivative discriminant \( D \) when performing the second derivative test to correctly classify the critical points.

Explore how to determine local extrema of functions of two variables using the second derivative test. This problem analyzes a function that attains a local maximum at a critical point, involving the Hessian matrix and discriminant calculations. Suitable for advanced undergraduate students in mathematics.

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