Providing an informal interpretation of Lagrange multiplier method (under 30 words)

Question

So the assignment is to provide an informal interpretation of Lagrange multiplier method (under 30 words and without math formulas). The main idea is to show intuition on why the method works and then state the insight in common and concise words to convince someone with sufficient math education that you understand why it works.

So far I am getting this

"Lagrange multiplier gives us normal vectors of constraint and level curves inline with each other of the extreme value with respect to the constraint".

Can someone comment if what I am saying makes sense to the assignment ?

Answer 1

60 words.

If the restriction of a function to an hypersurface admits a maximum at a point, then its derivative vanishes on the tangent hyperplane of the hypersurface at this point. If a linear form (the derivative of the function) vanishes on the kernel on another one (the tangent hyperplane), then they are proportional. The Lagrange multiplier is the coefficient of proportionality.

Answer 2

When miniminzing a function under an equality constraint, the gradient must be orthogonal to the constraint surface, i.e. equal to the gradient of the constraint function times a multiplier.