First-order condition constrained extremum problem

Introduction: A constrained extremum problem is given by
  • Optimize $z(x,y)$
  • Subject to $g(x,y)=k$
  • Where $x \in D_1$, $y \in D_2$
Theorem: An extremum location $(c,d)$, where $c \in D_1$ and $d \in D_2$ that is not a boundary point, satisfies the following system of equations:
  • $\dfrac{z'_x(x,y)}{z'_y(x,y)}=\dfrac{g'_x(x,y)}{g'_y(x,y)}$
  • $g(x,y)=k$

Remark: Do not forget to check whether the extremum is a minimum or a maximum.