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I know that the minimum or maximum point is achieved when the gradient in the constraint function is parallel to the gradient on the $f$ function. But why the Lambda is called the Lagrange multiplier?

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Not a direct answer, but this may help you remember the content of the theorem: A function will have a critical point at a set of constraints when its derivative is a linear combination of the derivatives of the constraints. – Lepidopterist Mar 18 '13 at 19:10
Are you asking for intuition or the naming origin? – copper.hat Mar 18 '13 at 19:12
I think that the terminology comes from statics, when the $f$ function is potential energy and the mechanical constraint is smooth (meaning that friction doesn't do work). The function $f$ has a critical point exactly when energy is stationary, that is, at equilibrium positions. In this case $\nabla f$ is a force and $\nabla g$, the gradient of the constaint function, is the force exerted by the constraint to compensate for $\nabla f$. The Lagrange multiplier measures the amount of stress the constraint is subjected to. – Giuseppe Negro Mar 18 '13 at 19:16

Lagrange was the one who was first involved in calculus of variations and therefore people tend to call many things in calculus of variations after Lagrange (such as Lagrangian dynamics, Lagrange-Euler equations, Lagrange multipliers, etc...).

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