# What is the intuition behind the Lagrange multiplier?

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