
Lagrange Multipliers - why is the gradient perpendicular to boundary curve at an extrema on the boundary curve
I am referring to this video at timestamp 10:38. He says that if there is a maximum on the curve, the curve should be flat. The gradient of f should be perpendicular to f at the point. Since, in an earlier part of the video, he said that the boundary curve is not a level curve (but the constraint is)? I don't get why the gradient of f should be perpendicular. Any help or intuitive explanation, please.