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Lagrange multipliers

Lagrange's method finds extreme points of $f$ when $(x,y)$ is constrained to a curve $g(x,y)=0$ . At the extreme point the level curve of $f$ is tangent to the constraint, so the gradients are parallel.

\nabla f = \lambda\nabla g

the Lagrange condition, together with

g=0

Symbols

$\lambda$	Lagrange multiplier
$g(x,y)=0$	the constraint

Example

Maximize $f=xy$ subject to $x+y=10$ : $\nabla f=(y,x)=\lambda(1,1) \Rightarrow x=y$ , which with $x+y=10$ gives $x=y=5$ and $f=25$ .

Lagrange's method gives candidates for extreme points - check whether they are a max or min by comparing function values.

Practise partial derivatives for free →

← The second derivative test · Double integral →

Part of Multivariable Calculus: Partial derivatives.