The method of Lagrange multipliers
In and , when optimizing a function subject to the equation , we use the fact that ( is called the Lagrange multiplier)
Find the maximum area for a rectangle inscribed in the ellipse
Use Lagrange multipliers to find the extreme values of the function subject to the given constraint.
;
;
;
Use Lagrange multipliers to find the points on the cone that are closest to the point .
When optimizing a function
subject to two constraints
and
,
we use
Find the extreme values of
subject to both constraints.
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,