High number_ Chapter 2 differential calculus of multivariate functions__ Method of solving conditional extremum__ Lagrange multiplier method

The general formulation of conditional extremum of binary function is :    In terms of constraints   φ(x,  y) = 0, Seeking function z = f(x,  y) The minimum or maximum of .

Lagrange multiplier method ：   Let's have a function of two variables f (x, y) and   φ(x,  y) There are continuous partial derivatives in the region under consideration , And   φᵪ（x,  y）,    φᵧ(x, y) Different at the same time 0. 

Make L(x,  y) = f(x,  y) +  λφ(x, y).                       （1）

Where the constant   λ It is called Lagrange multiplier , L(x, y) It's called a Lagrange function .  seek L Two partial derivatives of , And establish equations

If the function z = f（x, y） In terms of constraints   φ (x, y)=0 The extreme point under is (x₀, y₀), There is  λ₀, bring   x₀,  y₀, 

λ₀  It's the solution of the system of equations .

Lagrange multiplication gives a general method for finding conditional extremum , Its step is ：

1. Write the Lagrange function according to the objective function and constraints （1）
2. Establish equations
3. Find all the solutions of the equations , If λ₀,  x₀, y₀ It's the solution of the system of equations , The point of （x₀,  y₀) Is the suspicious extreme point ,
4. Judgment point (x₀,   y₀) Is it an extreme point .