Lagrange Multipliers With Two Parameters

     The method of Lagrange multipliers can also be applied in situations with more than one constraint equation. Suppose we wish to locate an extremum of a function defined by subject to constraints and where and are also differentiable and and are not parallel. By generalizing Lagrange's theorem, it can be shown that if is the desired extremum, then there are numbers and such that  



and  


As in the case of one constraint, we proceed by first solving this system of equations simultaneously to find and then evaluating at each solution and comparing to find the assumed extremum.

Example (Lagrange Multipliers With Two Parameters) Use the method of Lagrange multipliers to find the required extrema for the two given constraints.

(a)  Find the maximum value of the function



on the curve of intersection of the plane and the cylinder

    Solution. We maximize the function



subject to the constraints



The Lagrange condition is so we solve the equations



Putting we get so Similarly, we have Substitution yields and so Then and so we have The corresponding values of are



Therefore the maximum of on the given curve is

(b) Find the maximum of subject to and

    Solution. We need to solve the system where and Therefore, we need to solve the system




            






which is










The solutions are



The maximum is




(c)  Find the minimum of subject to and

    Solution. We want to minimize subject to the side conditions



We form  


The conditions are

  
  
  
  
  
  
  
  
  
  
  
The first and third conditions give



so the second condition becomes We then have



The solution to this system is Therefore, the minimum is  


(d) Use Lagrange multipliers to find the point on the line of intersection of the planes and that is closest to the origin.

    Solution. We want to minimize subject to the side conditions



We form  


The conditions are

  
  
  
  
  
  
  
  
  
  
  
The second and third conditions give



so the first condition becomes We then have



The last two equations may be written as and Substitution of these values into the first equation gives Consequently, and The desired point is therefore,

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