The image contains several mathematical problems related to finding maximum and minimum values, as well as extreme values of functions, constrained by various conditions. Here is a brief overview of the problems:

Problems Overview

1. Problem 16: Find the extreme values of $r^2$ where $r^2 = \frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4}$, subject to the constraint: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, \quad lx + my + nz = 0.$

2. Problem 17: Find the maximum and minimum values of $x^2 + y^2 + z^2$ subject to the constraints: $\frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} = 1, \quad z = x + y.$

3. Problem 18: Find the maximum and minimum values of $r^2$ where $r^2 = ax^2 + by^2 + cz^2$, subject to the constraints: $x^2 + y^2 + z^2 = 1, \quad lx + my + nz = 0.$

4. Problem 19: Find the maximum and minimum radii vectors of the section of the surface: $(x^2 + y^2 + z^2) l^2 + m^2 y^2 + n^2 z^2 = 1,$ made by the plane $ax + by + cz = 0$.

5. Problem 20: Find the lengths of the axes of the section of the ellipsoid: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$ by the plane $lx + my + nz = 0$.

6. Problem 21: Find the maximum and minimum radii vectors of the section of the surface: $ax^2 + by^2 + cz^2 = 1,$ by the plane $lx + my + nz = 0$, or find the extreme value of $x^2 + y^2 + z^2$ subject to the conditions $ax^2 + by^2 + cz^2 = 1$ and $lx + my + nz = 0$.

Solutions Approach

For most of these problems, the method of Lagrange multipliers would be a suitable approach since they involve finding the extreme values of functions subject to given constraints.

Would you like a detailed solution to a specific problem, or an explanation of the general method to solve these types of problems?

Here are 5 related questions to expand your understanding:

1. How does the method of Lagrange multipliers help in finding the extrema of functions with constraints?
2. What is the geometric interpretation of finding the maximum and minimum values on an ellipsoid?
3. How would you solve for the extreme values of a function of three variables subject to two constraints?
4. Can these problems be visualized in three-dimensional space, and if so, how?
5. What role do partial derivatives play in solving these optimization problems?

Tip: When working with multiple constraints, ensure that the gradients of all functions involved are properly aligned for the use of Lagrange multipliers.