Lagrange Multipliers in One Dimension
Lagrange Multipliers in One Dimension
This Demonstration shows how Lagrange multipliers work in one dimension. The 1D solution is trivial, as the solution is given by the constraint. Nevertheless, the 1D problem exhibits some essential features of the situation. The 2D problem, which is more difficult to visualize, is treated in another Demonstration.
The method of Lagrange multipliers takes the problem of finding the extreme value of a function subject to a constraint and replaces it with the problem of solving the equation subject to the same constraint. In one dimension, there is not much to this; we simply choose so that and to satisfy . The visual effect of this choice of (right panel) is to adjust the blue tangent so that it is parallel to the red tangent, or, equivalently, adjust the green tangent so that it is horizontal.
f(x)
R(x)=K
f'(x)-λR'(x)=0
x
0
R()=K
x
0
λ
f'()-λR'()=0
x
0
x
0
λ