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The given function is \( f(x, y, z) = 4y - z + 2 \).

To determine the nature of the level surfaces, we need to set the function equal to a constant, say \( c \), to represent the level surfaces:

\[
4y - z + 2 = c
\]

This simplifies to:

\[
z = 4y + 2 - c
\]

This is a linear equation in \( y \) and \( z \), meaning that for any value of \( c \), the level surface is a plane. Since the form of this equation remains similar regardless of the value of \( c \), these planes are parallel to one another.

Thus, the level surfaces represent **a family of parallel planes**.

The correct answer is: 

- The level surfaces are a family of parallel planes.

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

Here are 5 related questions for further exploration:
1. What are level surfaces in a 3D scalar field?
2. How do you identify the type of surface given a scalar function?
3. Can level surfaces represent something other than planes in different functions?
4. How does changing the constant \( c \) affect the position of the planes in space?
5. What is the geometric meaning of a level surface in physics?

**Tip:** Linear functions in terms of multiple variables often lead to planar level surfaces.

Given the function f(x, y, z) = 4y - z + 2, determine the nature of the level surfaces.

Learn how to identify the nature of level surfaces for the function f(x, y, z) = 4y - z + 2. This example focuses on linear functions and their corresponding planar level surfaces. Suitable for students studying multivariable calculus.

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