Lagrange Multiplier Summary

The method of Lagrange Multipliers can be summarized in one single formula:

\huge\boxed{\nabla f=\lambda\nabla g}

where f is the function to be optimized and g=0 is the constraint.

Example

Let’s illustrate this using the example in your notes (pg 40): Find the relative extrema of f(x,y)=12x-16y+50 subject to the constraint x^2+y^2=25.

Let g(x,y)=x^2+y^2-25 denote the constraint. According to the formula \boxed{\nabla f=\lambda\nabla g},

\begin{pmatrix}f_x\\f_y\end{pmatrix}=\lambda\begin{pmatrix}g_x\\g_y\end{pmatrix}

\begin{pmatrix}12\\-16\end{pmatrix}=\lambda\begin{pmatrix}2x\\2y\end{pmatrix}

Now we need to solve the simultaneous equations: 12=\lambda 2x, and -16=\lambda 2y, and the constraint x^2+y^2=25.

From this part onwards, the solving is identical to the part in the notes.

To understand Lagrange Multiplier intuitively, check out this nice post on Quora.

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