Intuição para os multiplicadores de Lagrange

Suppose that we would like to find the extreme values of some function $f(x_1,x_2,...,x_n)$ subject to some constraint $g(x_1,x_2,...,x_n)=0$.

To do so, let us draw level surfaces for the function $f$: these are surfaces such that every point on the surface has the same value of $f$.

The level surface might intersect the constraint surface in some region or curve. If this is the case, then changing the value of $f$ by a small amount will result in a new level curve, which also intersects the constraint surface in some region or curve. Thus, this level curve is not representative of an extreme value of $f$. This is similar to the case of a single variable function with $f'(x)\ne 0$ for some value $x$. Since the values of $f$ can go either way, $f$ does not have a maximum or minimum at $x$.

However, the other case is that the level surface is tangent to the constraint curve in a single point. In this case, changing the value for the level curve by even a small amount in one direction leads to no more intersection between the level surface and the constraint curve. Thus, $f$ achieves either a maximum or minimum at this point.

For the two surfaces to be tangent, their normal vectors must be parallel. Thus, one normal vector is the other vector multiplied by a scalar, known as a Lagrange multiplier.