Given the function \( f(x,y) = (x-y)(1-xy) \), let's explore its structure and properties:

### 1. **Expression Breakdown**:
   - The function is defined as a product of two terms:
     - The first term is \( (x-y) \).
     - The second term is \( (1-xy) \).
  
### 2. **Possible Simplifications**:
   - The function is already in its simplest form since the terms \( (x-y) \) and \( (1-xy) \) cannot be factored further.

### 3. **Partial Derivatives**:
   - To understand how \( f(x,y) \) behaves with respect to each variable, we can calculate the partial derivatives.
   - **Partial derivative with respect to \( x \)**:
     \[
     \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left[ (x-y)(1-xy) \right]
     \]
     Using the product rule:
     \[
     \frac{\partial f}{\partial x} = (1-xy) \cdot \frac{\partial}{\partial x}(x-y) + (x-y) \cdot \frac{\partial}{\partial x}(1-xy)
     \]
     Simplifying further:
     \[
     \frac{\partial f}{\partial x} = (1-xy)(1) + (x-y)(-y)
     \]
     So,
     \[
     \frac{\partial f}{\partial x} = 1 - xy - y(x-y)
     \]

   - **Partial derivative with respect to \( y \)**:
     \[
     \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left[ (x-y)(1-xy) \right]
     \]
     Again, using the product rule:
     \[
     \frac{\partial f}{\partial y} = (1-xy) \cdot \frac{\partial}{\partial y}(x-y) + (x-y) \cdot \frac{\partial}{\partial y}(1-xy)
     \]
     Simplifying further:
     \[
     \frac{\partial f}{\partial y} = (1-xy)(-1) + (x-y)(-x)
     \]
     So,
     \[
     \frac{\partial f}{\partial y} = -1 + xy - x(x-y)
     \]

### 4. **Critical Points**:
   - To find critical points, set both partial derivatives equal to zero and solve for \( x \) and \( y \).

### 5. **Second Derivatives**:
   - To analyze the concavity and determine the nature of the critical points (minima, maxima, or saddle points), we would calculate the second partial derivatives and use the Hessian matrix.

Would you like to explore any specific properties of this function, or do you have any particular questions in mind?

### Related Questions:
1. What are the critical points of the function \( f(x, y) \)?
2. How can we determine the nature of the critical points using the Hessian matrix?
3. What is the physical or geometric interpretation of the function \( f(x,y) \)?
4. How does the function behave along specific lines, such as \( y = x \) or \( y = -x \)?
5. Can the function be integrated over a specific domain, like a square or a circle?
6. What are the global maximum and minimum values of \( f(x,y) \)?
7. How does the function behave when \( x \) or \( y \) approach infinity?
8. What happens to \( f(x,y) \) when \( x \) and \( y \) are close to zero?

### Tip:
When analyzing multivariable functions, always start by finding the critical points through partial derivatives and then determine their nature using the second derivative test.

Understand the function f(x, y) = (x-y)(1-xy) through its structure, partial derivatives, critical points, and second derivative test. Learn how to analyze the behavior of this multivariable function. Suitable for advanced undergraduate students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation