Editing Lagrange multiplier (section)

=== Example 2 ===
[[Image:Lagrange very simple-1b.svg|thumb|right|300px|Illustration of the constrained optimization problem&nbsp;'''2''']]

Now we modify the objective function of Example&nbsp;'''1''' so that we minimize <math>\ f(x,y) = (x + y)^2\ </math> instead of <math>\ f(x,y) = x + y\ ,</math> again along the circle <math>\ g(x,y)=x^2+y^2-1 = 0 ~.</math> Now the level sets of <math>f</math> are still lines of slope −1, and the points on the circle tangent to these level sets are again <math>\ (\sqrt{2}/2,\sqrt{2}/2)\ </math> and <math>\ (-\sqrt{2}/2,-\sqrt{2}/2) ~.</math> These tangency points are maxima of <math>\ f ~.</math>

On the other hand, the minima occur on the level set for <math>\ f = 0\ </math> (since by its construction <math>\ f\ </math> cannot take negative values), at <math>\ (\sqrt{2}/2,-\sqrt{2}/2)\ </math> and <math>\ (-\sqrt{2}/2, \sqrt{2}/2)\ ,</math> where the level curves of <math>\ f\ </math> are not tangent to the constraint. The condition that <math>\ \nabla_{x,y,\lambda}\left(f(x,y)+\lambda\cdot g(x,y) \right)=0\ </math> correctly identifies all four points as extrema; the minima are characterized in by <math>\ \lambda =0\ </math> and the maxima by <math>\ \lambda = -2 ~.</math>