- Extrema of
subject to constraint satisfy - Extrema of
subject to constraints satisfy - Extrema
is a consequence of: - The Gradient of
being normal to the level curve , everywhere - The gradient of
along the boundary imposed by the constraint is also normal to the level curve , wherever the Parameterization of along the boundary ( ; single-variable calculus) would have a zero derivative, i.e. where there are possible Extrema. along the boundary is normal to the level curve at possible extrema because its parameterization is flat.
- The Gradient of
- See here for a great video explanation
Examples
Example 1
Find the points on surface
that are closest to the origin. Let be the distance from a point to the origin Let be the proxy to the distance function that we will use to solve this problem, because its extrema exactly correspond to that of . is our objective function Let be our constraint function Goal: find points where and Find Gradients: Solve this system:
Blah blah blah… You do a bunch of different cases and get the following points: You can plug each of these into and you’ll find and