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The problem asks to find the point on the paraboloid \( z = x^2 + y^2 + 2 \) that is closest to the point \( P = (2, 2, 2) \).

This type of problem can be approached by minimizing the distance between a point on the surface of the paraboloid and the given point \( P \). Let me walk you through the solution step by step:

### Step 1: Distance Formula

The distance \( D \) between any point on the paraboloid \( (x, y, z) \) and the point \( P(2, 2, 2) \) is given by:

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

Since \( z = x^2 + y^2 + 2 \) on the surface, we can substitute \( z \) into the distance function:

\[
D(x, y) = \sqrt{(x - 2)^2 + (y - 2)^2 + (x^2 + y^2 + 2 - 2)^2}
\]
\[
D(x, y) = \sqrt{(x - 2)^2 + (y - 2)^2 + (x^2 + y^2)^2}
\]

### Step 2: Minimize the Distance

To find the minimum distance, we need to minimize \( D(x, y) \), or equivalently, minimize the square of the distance (since minimizing \( D^2(x, y) \) will also minimize \( D(x, y) \)):

\[
D^2(x, y) = (x - 2)^2 + (y - 2)^2 + (x^2 + y^2)^2
\]

Now, we take partial derivatives of \( D^2(x, y) \) with respect to \( x \) and \( y \), and solve for the critical points.

### Step 3: Solving the Partial Derivatives

I will now calculate the derivatives and find the critical points to determine the closest point on the paraboloid.The critical points calculated are quite complex due to the nature of the expressions. However, in practical terms, these results suggest that the closest point can be found through numerical approximation.

Let me simplify and compute these numerically to identify the exact values for the point closest to \( P = (2, 2, 2) \).The numerical approximation of the critical points gives:

1. Real point: \( (x, y) \approx (0.689, 0.689) \)
2. Complex points: These involve imaginary components, so they are not physically relevant in this context.

Therefore, the closest point on the paraboloid \( z = x^2 + y^2 + 2 \) to the point \( P = (2, 2, 2) \) is approximately \( (0.689, 0.689) \).

### Step 4: Finding the \( z \)-coordinate
Now, we can substitute \( x = 0.689 \) and \( y = 0.689 \) into the equation of the paraboloid to find \( z \):

\[
z = x^2 + y^2 + 2
\]

Let me calculate the \( z \)-coordinate.The \( z \)-coordinate corresponding to the point \( (x, y) \approx (0.689, 0.689) \) is approximately \( z \approx 2.951 \).

### Conclusion
The point on the paraboloid \( z = x^2 + y^2 + 2 \) that is closest to \( P = (2, 2, 2) \) is approximately:

\[
(0.689, 0.689, 2.951)
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How do we use Lagrange multipliers for minimizing distances on surfaces?
2. How can we visualize the paraboloid and the point \( P \) in 3D space?
3. What happens if the point \( P \) were located at a different height, say \( z = 5 \)?
4. How would the process change if the surface equation was different (e.g., an ellipsoid)?
5. Can we compute the distance numerically without solving analytically?

**Tip:** When dealing with optimization problems on surfaces, minimizing the square of the distance often simplifies calculations since it avoids handling square roots.

Hallar el punto del paraboloide z = x^2 + y^2 + 2 más cercano del punto P = (2,2,2).

This page walks through solving the problem of finding the point on the paraboloid z = x^2 + y^2 + 2 closest to the point P = (2,2,2). Using optimization techniques and the distance formula, we minimize the square of the distance function to find the closest point. Partial derivatives and numerical methods are used in the process.

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