Consider a function of three independent variables \( f(x,y,z) \) . For the function \( f \) to be an extreme we have $$ df=0. $$ A necessary and sufficient condition is $$ \frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, $$ due to $$ df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. $$ In many problems the variables \( x,y,z \) are often subject to constraints (such as those above for the margin) so that they are no longer all independent. It is possible at least in principle to use each constraint to eliminate one variable and to proceed with a new and smaller set of independent varables.
The use of so-called Lagrangian multipliers is an alternative technique when the elimination of variables is incovenient or undesirable. Assume that we have an equation of constraint on the variables \( x,y,z \) $$ \phi(x,y,z) = 0, $$ resulting in $$ d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. $$ Now we cannot set anymore $$ \frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, $$ if \( df=0 \) is wanted because there are now only two independent variables! Assume \( x \) and \( y \) are the independent variables. Then \( dz \) is no longer arbitrary.