Multivariable Optimization (Unconstrained & Constrained) + SOC (Hessian Idea)

1) Unconstrained optimization (two variables)

Maximize/minimize:

1.1 First-order conditions (FOC)

Solutions are critical points.

2) Second-order conditions (SOC): Hessian (CUET-level)

Second derivatives:

Hessian matrix:

Determinant of Hessian:

Classification (for two variables):

• Local minimum if $D>0$ and $f_{xx}>0$
• Local maximum if $D>0$ and $f_{xx}<0$
• Saddle point if $D<0$
• If $D=0$: inconclusive

3) Constrained optimization: Lagrange method (practice-ready)

Maximize $f(x,y)$ subject to:

Lagrangian:

FOCs: