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Wikipedia gives the following illustration for the method of Lagrange multipliers. In this case, $d_1$ is the highest-valued contour line, so clearly the method works.

enter image description here

But what if $d_2$ or $d_3$ were higher than $d_1$? Wouldn't the method fail? If so, what conditions must $f$ satisfy for the method of Lagrange multipliers to apply? If not, why not?

Tyler
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  • For non-convex optimization problems, there may be Lagrange points that are local maximums, local minimums, and saddle points, there's no guarantee that a Lagrange point is a global maximum. – Brian Borchers Jan 10 '23 at 22:30

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