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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.

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?

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1 Answer

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The Lagrange Multiplier is a piece of the process in Constrained Optimization.

In single variable calculus, when you optimize a function, you do not know whether you are finding a local minimum or maximum, but simply a stationary point. You use the second derivative test to determine that.

The same applies in this case, it is perfectly reasonable for d2 and d3 to be higher, in that case the Lagrange multiplier helped you find a local minimum! You can use a second derivative test to determine it.

Note in higher than 2 dimensions you can come up with other answers than a + or - concavity, such as a saddle point.

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