Could someone explain the Lagrangian Method?

Question

I understand the method, technically, but what is actually going on?

We set the gradient of the function equal to the gradient of the constraint (multiplied by a constant), and in doing so, we find local extrema. I'm not sure what is actually going on here though, could someone provide me with some intuition?

Answer 1

Tony Piccola's answer was the following website.

http://www.the-idea-shop.com/article/215/understanding-why-the-method-of-lagrange-multipliers-works

I found this very intuitive, so hopefully it helps others.

Answer 2

If the function $f$ has a local extremal point contrained on a surface $g=0$, then it means that the derivatives of $f$ along the tangent directions are zero. This means that the gradient $\nabla f$ has no tangent component, hence it is normal to the surface $g=0$. But the direction of the normal to a surface $g=0$ is the gradient $\nabla g$. So we have that $\nabla f$ is parallel to $\nabla g$.