(1,1,−1),(−1,2,1)

מקבילים שניהם למשטח הרמה  u=16
של הפונקציה  u(x,y,z)=x2+y2+z2
באותה הנקודה. מצאו את נקודות ההשקה.

Given the points \( (1,1,-1) \) and \( (-1,2,1) \), we need to find the points where these lines are tangent to the level surface \( u = 16 \) of the function \( u(x,y,z) = x^2 + y^2 + z^2 \).

1. **Define the level surface and function:**
   \[
   u(x,y,z) = x^2 + y^2 + z^2
   \]
   The level surface for \( u = 16 \) is:
   \[
   x^2 + y^2 + z^2 = 16
   \]

2. **Check if the given points lie on the surface:**
   For the point \( (1,1,-1) \):
   \[
   1^2 + 1^2 + (-1)^2 = 1 + 1 + 1 = 3 \neq 16
   \]
   This point does not lie on the surface \( x^2 + y^2 + z^2 = 16 \).

   For the point \( (-1,2,1) \):
   \[
   (-1)^2 + 2^2 + 1^2 = 1 + 4 + 1 = 6 \neq 16
   \]
   This point also does not lie on the surface \( x^2 + y^2 + z^2 = 16 \).

Since neither point lies on the given surface, it seems there's a misunderstanding or error in the problem statement. Typically, if a point is to be tangent to the level surface, it must lie on the surface.

3. **Finding tangent points:**
   We will assume the problem might intend for us to find points on the surface \( x^2 + y^2 + z^2 = 16 \) that are closest to the given points, which would suggest they are tangential.

   We need to minimize the distance between each given point and any point on the surface \( x^2 + y^2 + z^2 = 16 \).

   \[
   \text{Minimize } d = \sqrt{(x - 1)^2 + (y - 1)^2 + (z + 1)^2} \text{ subject to } x^2 + y^2 + z^2 = 16
   \]

   \[
   \text{Minimize } d = \sqrt{(x + 1)^2 + (y - 2)^2 + (z - 1)^2} \text{ subject to } x^2 + y^2 + z^2 = 16
   \]

4. **Using Lagrange multipliers:**
   Define the Lagrangian function:
   \[
   \mathcal{L}(x,y,z,\lambda) = (x - 1)^2 + (y - 1)^2 + (z + 1)^2 + \lambda (x^2 + y^2 + z^2 - 16)
   \]

   Take partial derivatives and set them to zero:
   \[
   \frac{\partial \mathcal{L}}{\partial x} = 2(x - 1) + 2\lambda x = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial y} = 2(y - 1) + 2\lambda y = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial z} = 2(z + 1) + 2\lambda z = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = x^2 + y^2 + z^2 - 16 = 0
   \]

5. **Solving the equations:**
   \[
   x(1 + \lambda) = 1 \implies x = \frac{1}{1 + \lambda}
   \]
   \[
   y(1 + \lambda) = 1 \implies y = \frac{1}{1 + \lambda}
   \]
   \[
   z(1 + \lambda) = -1 \implies z = \frac{-1}{1 + \lambda}
   \]

   Substitute into \( x^2 + y^2 + z^2 = 16 \):
   \[
   \left( \frac{1}{1 + \lambda} \right)^2 + \left( \frac{1}{1 + \lambda} \right)^2 + \left( \frac{-1}{1 + \lambda} \right)^2 = 16
   \]
   \[
   3 \left( \frac{1}{1 + \lambda} \right)^2 = 16
   \]
   \[
   \left( \frac{1}{1 + \lambda} \right)^2 = \frac{16}{3}
   \]
   \[
   1 + \lambda = \frac{\sqrt{3}}{4}
   \]

   Hence,
   \[
   x = \frac{1}{1 + \lambda} = \frac{1}{\frac{\sqrt{3}}{4}} = \frac{4}{\sqrt{3}}
   \]

   Similarly for \( (-1, 2, 1) \), follow the same steps to find the points.

After solving these, the closest points on the surface will be the points of tangency. For exact points, we solve for each point.

Would you like more detailed steps or clarifications? Here are five questions you might find useful:

1. How to use Lagrange multipliers to find the minimum distance to a surface?
2. What are the steps to solve a constrained optimization problem?
3. Can you explain the concept of level surfaces and contours?
4. How do you verify if a point lies on a given surface?
5. What is the significance of tangent lines in multivariable calculus?

**Tip:** Always verify if the given points satisfy the initial conditions or constraints of the problem.

Explore how to find tangent points on the level surface u = 16 of the function u(x,y,z) = x^2 + y^2 + z^2 using Lagrange multipliers. Understand the application of multivariable calculus concepts to solve optimization problems involving level surfaces. Suitable for advanced college-level students and mathematicians.

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