**Finding the Maximum or Minimum Value of a Function of Three Variables**

When dealing with a function of three variables, the task is to find the maximum or minimum value of the function. There are various methods available to solve this problem, each with its own advantages and disadvantages. Let's discuss these methods in detail:

**1. Newton Method:**
The Newton method, also known as Newton's method or Newton-Raphson method, is typically used for finding the roots of a function. It is not directly applicable to finding the maximum or minimum value of a function of three variables. Therefore, the Newton method is not the correct answer for this question.

**2. Euler Method:**
The Euler method, also known as the Euler's approximation method, is used to numerically solve ordinary differential equations. It is not applicable for finding the maximum or minimum value of a function of three variables. Hence, the Euler method is also not the correct choice for this question.

**3. Lagrange Method:**
The Lagrange method, also known as the method of Lagrange multipliers, is a powerful technique used to find the maximum or minimum value of a function subject to one or more constraints. This method involves finding the critical points of the function by setting the partial derivatives equal to zero and then solving a system of equations. However, the Lagrange method is not specifically designed for functions of three variables. It can be extended to handle functions of multiple variables, including three variables. Therefore, the Lagrange method is a valid option for finding the maximum or minimum value of a function of three variables.

**4. Bessel Method:**
The Bessel method, named after the mathematician Friedrich Bessel, is primarily used to solve differential equations in cylindrical coordinate systems. It is not applicable for finding the maximum or minimum value of a function of three variables. Thus, the Bessel method is not the correct answer for this question.

**Conclusion:**
Out of the given options, the most appropriate method for finding the maximum or minimum value of a function of three variables is the Lagrange method. Although not specifically designed for three variables, it can be extended to handle functions of multiple variables effectively. Hence, option (b) Lagrange method is the correct choice for this question in the Railways category.