How do you maximize and minimize #f(x,y)=x-siny# constrained to #0<=x+y<=1#?
1 Answer
Write a Lagrange function with 2 multipliers and 2 slack variables.
Compute the partial derivatives.
Solve the system of equations.
Explanation:
For the constraint,
We add two slack variables into two constraint functions:
Note: squaring the slack variables assures that the constraint is enforced by disallowing negative values.
We can write the Lagrange function:
Compute the partial derivatives:
Set the partial derivatives equal to 0 and the solve as a system of nonlinear equations:
Please observe that the extrema are located at the points are where equations [3] and [4] are satisfied by
Because u and v are not forced to be zero, we can subtract equation [2] from equation [1]
Using equation [5.1] , we obtain the value for x:
This gives the function's minimum:
For the maximum, we have the condition where
Equation [6.2] gives us the value for x: