https://file.aiquickdraw.com/mathbot/file/1729227706377ygvvzkdl.jpg

The image presents a 3D graph of a function \( f \), and the task is to determine the signs of the partial derivatives of \( f \) at specific points. Let's break it down.

### Key Details:
- The graph shows a surface with axes labeled \( X \), \( Y \), and \( Z \). The surface appears to have a saddle shape.
- There are two points on the graph marked as \( (x_0, y_0) \) and \( (-x_0, y_0) \).
- Two partial derivatives are being questioned:
  - (a) \( f_{xy}(x_0, y_0) \)
  - (b) \( f_{xy}(-x_0, y_0) \)

### Partial Derivative Signs:

1. **Partial Derivative \( f_{xy}(x_0, y_0) \):**
   - \( f_{xy}(x_0, y_0) \) represents the mixed partial derivative with respect to \( x \) and \( y \) at the point \( (x_0, y_0) \).
   - Looking at the surface around \( (x_0, y_0) \), the surface seems to have a positive slope in the \( x \)-direction while the slope in the \( y \)-direction seems to vary. Therefore, the mixed derivative is likely **negative** at this point, as indicated in the diagram (blue).

2. **Partial Derivative \( f_{xy}(-x_0, y_0) \):**
   - \( f_{xy}(-x_0, y_0) \) represents the mixed partial derivative at the point \( (-x_0, y_0) \), and visually, the slope in the \( x \)-direction seems to be negative while the slope in the \( y \)-direction appears to vary as well.
   - Based on the surface structure, this derivative is likely **positive**, as the slopes behave oppositely around the point \( (-x_0, y_0) \).

This aligns with the choices shown in the image. Specifically:
- (a) The sign of \( f_{xy}(x_0, y_0) \) is **negative**.
- (b) The sign of \( f_{xy}(-x_0, y_0) \) is **positive**.

Would you like more details on the interpretation of partial derivatives or how these signs are determined?

### 5 Related Questions:
1. What is the geometric meaning of partial derivatives in a 3D surface?
2. How do mixed partial derivatives like \( f_{xy} \) influence the shape of a surface?
3. What are the conditions under which a mixed partial derivative could be zero?
4. How can one visualize the impact of changing one variable on the slope of another in a multi-variable function?
5. What does the saddle shape of the surface imply about the nature of the function at certain points?

### Tip:
Mixed partial derivatives often help in understanding the interaction between two directions of change, and they are essential in studying the curvature and concavity of surfaces.

Determine the signs of the partial derivatives for the function f whose graph is shown below.

Explore how to determine the signs of partial derivatives for a given 3D surface function. The problem covers mixed partial derivatives, their interpretation in multivariable calculus, and the implications of surface curvature. Suitable for advanced high school or undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation