Skip to content
Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Lagrange Multipliers

A method for maximizing or minimizing a general function f(x,y,z)f(x, y, z) subject to a constraint (or side condition) of the form g(x,y,z)=kg(x, y, z) = k.

Theorem

Suppose f,gC1f,g \in C^1 and g0\nabla g \neq 0. Then the extremum(s) of f(x,y)f(x,y) subjected to g(x,y)=kg(x,y) = k are included in each of:

det(f,g)(x,y)=0    and    g(x,y)=0\det \frac{\partial (f,g)}{\partial (x,y)} = 0\;\;\text{and}\;\;g(x,y) = 0 f(x,y)=λg(x,y)    and    g(x,y)=0\nabla f(x,y) = \lambda \nabla g(x,y) \;\;\text{and}\;\; g(x,y) = 0

When there are nn constraint, the system of equations becomes:

The system of equations is{f=λ1g1+λ2g2++λngng1(x,y)=k1g2(x,y)=k2gn(x,y)=kn\text{The system of equations is} \begin{cases} \nabla f = \lambda_1 \nabla g_1 + \lambda_2 \nabla g_2 + \cdots + \lambda_n \nabla g_n \\ g_1(x,y) = k_1 \\ g_2(x,y) = k_2 \\ \vdots \\ g_n(x,y) = k_n \end{cases}