Visually and geometrically, I can see how this makes sense, but is there a more formal explanation of why this is true? With only an understanding of the geometric picture, I don't understand why the Lagrange multiplier method doesn't always provide all of points at which the function has an absolute maximum or absolute minimum, specifically in the case when the constraint curve closes in on itself.

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    $\begingroup$ Can you please clarify the question? Something like this? First, you show that a point is stationary iff it's stationary wrt the restriction of the map to the mentioned tangent line in said point, which is the case iff the derivative in this direction vanishes. But the derivative in this direction is eactly the inner product of the gradient with the direction, thus the gradient is orthogonal to this direction, the direction of the tangent line to the constraint curve at the point (which is stationary). $\endgroup$
    – Matija
    Oct 23, 2022 at 20:56


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