Functions of several variables: Lagrange multipliers
The method of Lagrange multipliers (examples)
Often you want to find extreme values under constraints. The method of Lagrange multipliers can help. We first deal with introductory examples.
We discuss two methods:
1. Direct method
In the constraint, we can isolate the variable :2. Lagrange multipliers
The task to find the minimum of
The method of Lagrange multipliers appeared out of the blue, but could be better understood as follows.
The constraint equation represents a straight line. We can now consider the contour curves of the function : these are circles with the origin as centre. If a contour curve intersects the straight line in two points, then the corresponding level of cannot be an extreme value, because a slightly larger or slightly smaller value leads to curves that intersect the line with equation in two points, too. An extremum (in this case a minimum) can only occur if the straight line and the contour curve touch each other, i.e., when the line is a tangent line of the contour curve. But this means that we must find a point for which and for which the gradient of and the gradient of calculated in that point are a multiple of each other. The requirements