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i was able to work out this proof of the idea, but its only true for a function with two independent variables and one dependent variable.

Rn=the space that is the independent variables.

x[Rn]=x

C[Rn]=C=constant.

dx/d[Rn]=grad(x)*v; v is a unit vector

dC/d[Rn]=grad(c)*v

because C is held constant, dC/d[Rn]=0 everywhere.

because cos(pi/2)=0, **grad(C) is perpendicular to v.**

In order for extrema to exist, dx/d[Rn]=0. grad(x)*v is zero meaning **grad(x) is perpendicular to v.**

in the case Rn=R2:

both grad(x) and grad(c) are perpendiular to v, it means grad(x) must be parallel to grad(c).

That is the requirement given by the system

grad(x)=L*grad(c)

C[Rn]=C

where L is the scalar multiplier (upside down y).

but it seems as though this is only true for the R2 case. in 3 dimnensions, if both grad(x) and grad(c) are perpendicular to v, it doesnt necessarily mean grad(x) is parallel to grad(c). It seems like im missing something.

How do i extend this to more than two independent variables?

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# Lagrangian multipliers

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