Absolute extrema of a function on a constraint set (3 variables)

Scroll down to the bottom to view the interactive graph.

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.

To view the interactive graph:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.