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The problem asks to find the points on the hyperbolic cylinder \(x^2 - z^2 = 1\) that are closest to the point \((-1, 1, 1)\). It is structured as a constrained optimization problem, where the goal is to minimize the distance function \(f(x, y, z)\), subject to the constraint \(g(x, y, z) = 0\), where \(g(x, y, z)\) represents the equation of the hyperbolic cylinder.

### Step-by-step breakdown:
1. **Objective function \(f(x, y, z)\):** The function we want to minimize is the square of the Euclidean distance between any point \((x, y, z)\) and the point \((-1, 1, 1)\):
   \[
   f(x, y, z) = (x + 1)^2 + (y - 1)^2 + (z - 1)^2
   \]
   This is because the distance formula is \(\sqrt{(x + 1)^2 + (y - 1)^2 + (z - 1)^2}\), and to simplify calculations, we minimize the square of the distance instead of the distance itself.

2. **Constraint \(g(x, y, z)\):** The equation of the hyperbolic cylinder is:
   \[
   g(x, y, z) = x^2 - z^2 - 1 = 0
   \]
   This ensures that the points lie on the surface of the cylinder.

Thus, the correct choice for \(f(x, y, z)\) and \(g(x, y, z)\) is:
\[
f(x, y, z) = (x + 1)^2 + (y - 1)^2 + (z - 1)^2, \quad g(x, y, z) = x^2 - z^2 - 1
\]

This corresponds to the **last option** in the list.

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Do you need further clarification or details on solving the problem using Lagrange multipliers or other methods? Here are some related questions to expand on this:

1. How do you apply Lagrange multipliers to solve this minimization problem?
2. What is the geometric interpretation of the hyperbolic cylinder \(x^2 - z^2 = 1\)?
3. Why is it valid to minimize the square of the distance rather than the distance itself?
4. What are the necessary conditions for a minimum in a constrained optimization problem?
5. How can we visualize the problem in 3D space for better understanding?

**Tip:** When minimizing distance, it's often easier to minimize the square of the distance, as it avoids dealing with square roots and yields the same point of minimum.

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

This problem involves finding the closest points on the hyperbolic cylinder x^2 - z^2 = 1 to the point (-1, 1, 1) using multivariable calculus and the Lagrange multiplier method. It demonstrates optimization techniques in constrained environments, including the use of distance formulas and hyperbolic cylinder equations.

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