How does lagrange multipliers work? 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?