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Here we shows the extreme value of \[f(x,y,z) = x-y-z\] subject to the constraint \[g(x,y,z) = x^2+y^2+z-2 = 0.\] On the constraint set (the blue surface), the lowest value of \(f(x,y,z)\) one can attain is \(-2.5\). This happens when the level surface \(f(x,y,z) = C\) (the red surface) just touches the constraint set, which is precisely at \(C = -2.5\).
There is no maximum value of \(f\) on the given constraint set.
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