Find the extreme values of f subject to both constraints

f (x , y , z ) = z ; x^2 + y^2 = z^2 , x + y + z = 24

The Lagrangian is



with partial derivatives (set equal to zero) yielding



Subtracting the second equation from the first gives



So in the fourth and fifth equations, we have



There are then two critical points for at and .

Clearly, attains a local minimum at the first point of , and a local maximum at the second point of .