https://file.aiquickdraw.com/mathbot/file/172869504860315038yqv.jpg

To solve this problem, we need to verify whether a function \( z = f(x, y) \) exists that has the given partial derivatives. The partial derivatives are as follows:

\[
\frac{\partial f}{\partial x} = 3x^2y - 4x - 3
\]
\[
\frac{\partial f}{\partial y} = 4x^3y^2 + 8y^3
\]

### Step 1: Check for Mixed Partial Derivatives
To determine if such a function \( f(x, y) \) exists, we should check whether the mixed partial derivatives are equal. Specifically, we should compute:

1. \(\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right)\)
2. \(\frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right)\)

#### 1. Compute \(\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right)\)
\[
\frac{\partial}{\partial y} \left( 3x^2y - 4x - 3 \right) = 3x^2
\]

#### 2. Compute \(\frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right)\)
\[
\frac{\partial}{\partial x} \left( 4x^3y^2 + 8y^3 \right) = 12x^2y^2
\]

### Step 2: Compare the Mixed Partial Derivatives
We find that:
\[
\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = 3x^2
\]
\[
\frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = 12x^2y^2
\]

These are not equal. Therefore, **it is not possible to find a function \( f(x, y) \) with the given partial derivatives** because the mixed partial derivatives are not equal. This implies that no such function \( f(x, y) \) exists.

### Answer
The correct choice is:
**A. It is not possible to find such a function because \(\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = 3x^2\), which is not equal to \(\frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = 12x^2y^2\).**

---

Would you like further details or have any questions about this solution?

#### Related Questions:
1. What conditions are necessary for a function to have continuous partial derivatives?
2. How can one verify if a given function has mixed partial derivatives that are equal?
3. What theorem is typically used to determine the equality of mixed partial derivatives?
4. In what scenarios might partial derivatives be unequal, indicating no such function exists?
5. How can you integrate partial derivatives to find a function?

#### Tip:
Always check the mixed partial derivatives when determining if a given set of partial derivatives corresponds to an actual function. If they are not equal, it indicates that no such function exists.

Find a family of functions z=f(x,y) whose partial derivatives are given, or explain why this is impossible. Given partial derivatives: \( \frac{\partial f}{\partial x} = 3x^2y - 4x - 3 \), \( \frac{\partial f}{\partial y} = 4x^3y^2 + 8y^3 \).

Explore how to find a function z = f(x, y) based on its partial derivatives, with a detailed step-by-step solution. Learn about mixed partial derivatives and Clairaut's theorem, crucial for verifying the existence of such functions. Suitable for college-level 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