Mathematical knowledge arrangement: extremum & maximum, stagnation point, Lagrange multiplier

1 extremum

1.1 Stagnation point

• If the function z=f(x,y) At point There's an extreme value at , And partial derivative exists , that z=f(x,y) stay The partial derivative at must be 0, namely
• To satisfy The point of For the function z=f(x,y) Stop at , Also known as the equilibrium point
• z=f(x,y) Its extreme point must be the point where its stagnation point or partial derivative does not exist ; however The stagnation point is not necessarily the extreme point
  • such as z=f(x,y)=xy, stay (0,0) Obviously there is , That is to say (0,0) yes f(x,y) A stopping point of , however (0,0) It is not the extreme point of the function （ Just a saddle point ）

   1.1.1 The sufficient condition that the stagnation point is the extreme point

set up z=f(x,y) A point in the definition field Where there is a second-order continuous partial derivative , And

We remember  ,, be

• When Δ>0,A>0 when ,f(x,y) stay   spot There is a minimum at

• When Δ>0,A<0 when ,f(x,y) stay   spot There is a maximum at

• When Δ<0 when , f(x,y) stay   spot There is no extreme value at

 2 The most value

Find the function in the region D The general way to get the best value on is , First, find out the function in the region D Internal extremum , Is that the function is in D Compare the maximum value on the boundary of .

 

For some problems , We can know that the maximum value of a function exists , And in the area D Internal . If the function is in the region D The inner partial differential exists , And there is only one stopping point , Then this stagnation point is the most valuable

3 Conditional extremum

Seeking function z=f(x,y) Meet the conditions Conditional extremum of

3.1 The conditional extremum is transformed into ordinary extremum

From the conditional equation The solution is given in the paper y=g(x), Plug in z=f(x,y) in

3.2 Lagrange multiplier

（1） Write auxiliary functions

（2）x,y,λ The system of equations satisfied

（3） From the above equations (x,y) Is the point where the conditional extremum may be taken

seek m Meta function +n The two constraints are the same