# Number of Lagrange Multipliers

Suppose we are looking at the following:

$$\text{minimize} \ f(x) = x^2+y^2 \\ \text{subject to} \ \ x+y-2 \geq 0$$

Would there only be one Lagrange multiplier corresponding to the single constraint? Or would there be two Lagrange multipliers corresponding to $x$ and $y$?

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There would only be one. By the way, note that $x^2+y^2=(1/2)[(x+y)^2+(x-y)^2]\ge (1/2)(x+y)^2$, with equality when $x=y$. So the minimum is reached when $x=y=1$. –  André Nicolas May 9 '12 at 23:24
In general, each constraint gives you one Lagrange multiplier, no matter how many variables there are. –  Robert Israel May 9 '12 at 23:27