https://file.aiquickdraw.com/mathbot/file/1728433858078198x1bka.jpg

The problem presented in the image is asking to find the points on the hyperbolic cylinder \( x^2 - z^2 = 1 \) that are closest to the point \( (-1, 1, 1) \).

### Objective:
Minimize the distance between a point \((x, y, z)\) on the surface of the hyperbolic cylinder and the given point \((-1, 1, 1)\). 

The function \( f(x, y, z) \) represents the square of the distance between the points, and \( g(x, y, z) = 0 \) is the constraint for the hyperbolic cylinder.

### Distance Function:
The Euclidean distance between two points \( (x, y, z) \) and \( (-1, 1, 1) \) is:

\[
d = \sqrt{(x + 1)^2 + (y - 1)^2 + (z - 1)^2}
\]

However, for simplicity, we can minimize the square of this distance:

\[
f(x, y, z) = (x + 1)^2 + (y - 1)^2 + (z - 1)^2
\]

### Constraint:
The hyperbolic cylinder is defined by:

\[
g(x, z) = x^2 - z^2 - 1 = 0
\]

### Correct Choices:
The correct function for \( f \) and \( g \) is:

\[
f(x, y, z) = (x + 1)^2 + (y - 1)^2 + (z - 1)^2
\]
\[
g(x, z) = x^2 - z^2 - 1
\]

This matches the option that is highlighted in the image.

Would you like to proceed with solving this problem using Lagrange multipliers, or have further details about any part of the process?

### Related Questions:
1. How do you apply Lagrange multipliers to minimize functions with constraints?
2. Why do we minimize the squared distance rather than the actual distance?
3. What is the geometric interpretation of the hyperbolic cylinder \( x^2 - z^2 = 1 \)?
4. How can we find the gradient of \( f(x, y, z) \) and \( g(x, z) \) in this context?
5. Can this problem be generalized to other surfaces and points?

### Tip:
In optimization problems involving distance minimization with constraints, squaring the distance simplifies the derivatives while keeping the critical points unchanged.

Find the points on the hyperbolic cylinder x^2 - z^2 = 1 that are closest to the point (-1, 1, 1).

Learn how to find the points on the hyperbolic cylinder x^2 - z^2 = 1 that are closest to the point (-1, 1, 1). This problem involves minimizing the Euclidean distance under a constraint using Lagrange multipliers. Suitable for college-level students studying multivariable calculus.

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