Math Problem Statement

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

Solution

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

Key Details:

  • The graph shows a surface with axes labeled XX, YY, and ZZ. The surface appears to have a saddle shape.
  • There are two points on the graph marked as (x0,y0)(x_0, y_0) and (x0,y0)(-x_0, y_0).
  • Two partial derivatives are being questioned:
    • (a) fxy(x0,y0)f_{xy}(x_0, y_0)
    • (b) fxy(x0,y0)f_{xy}(-x_0, y_0)

Partial Derivative Signs:

  1. Partial Derivative fxy(x0,y0)f_{xy}(x_0, y_0):

    • fxy(x0,y0)f_{xy}(x_0, y_0) represents the mixed partial derivative with respect to xx and yy at the point (x0,y0)(x_0, y_0).
    • Looking at the surface around (x0,y0)(x_0, y_0), the surface seems to have a positive slope in the xx-direction while the slope in the yy-direction seems to vary. Therefore, the mixed derivative is likely negative at this point, as indicated in the diagram (blue).
  2. Partial Derivative fxy(x0,y0)f_{xy}(-x_0, y_0):

    • fxy(x0,y0)f_{xy}(-x_0, y_0) represents the mixed partial derivative at the point (x0,y0)(-x_0, y_0), and visually, the slope in the xx-direction seems to be negative while the slope in the yy-direction appears to vary as well.
    • Based on the surface structure, this derivative is likely positive, as the slopes behave oppositely around the point (x0,y0)(-x_0, y_0).

This aligns with the choices shown in the image. Specifically:

  • (a) The sign of fxy(x0,y0)f_{xy}(x_0, y_0) is negative.
  • (b) The sign of fxy(x0,y0)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 fxyf_{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.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Partial Derivatives
Mixed Partial Derivatives
3D Surface Analysis

Formulas

f_xy = ∂²f/∂x∂y (Mixed partial derivative formula)

Theorems

Clairaut's Theorem on Mixed Derivatives
Interpretation of Partial Derivatives in Multivariable Calculus

Suitable Grade Level

Undergraduate Calculus or Advanced High School (Grades 11-12)