Lagrange multipliers

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To optimize a function $f (x, y)$ subject to a constraint $g (x, y) = c$ , the gradients must be parallel at an extremum: $
abla f=\lambda\nabla g $. S o l v e t hi s v ec t or e q u a t i o n t o g e t h er w i t h t h eco n s t r ain t e q u a t i o n f or$ x $,$ y $, an d$ \lambda $. E a c h so l u t i o n p o in t i s a c an d i d a t e; e v a l u a t e$ f$ at all candidates to identify the maximum and minimum values.

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From open sources

Complete problems only, with worked solutions first. Show everything

#	Title	Difficulty
135122	With an inequality constraint …, the multiplier at a maximum point satisfies…	Deep
135123	When the constraint … is a strict inequality … at the minimum…	Deep
135125	Minimize … subject to the inequality constraint	Deep
135127	Minimize … subject to … and	Deep
135128	A wire … long is used to enclose _one or two_ squares…	Deep
135124	Minimize … subject to the inequality constraint	Deep