Unconstrained Multivariable Extrema

Finding Peaks and Pits

To find unconstrained extrema of $f(x,y)$, begin with First-Order Conditions: set $f_x = 0$ and $f_y = 0$. Their simultaneous solutions are critical points—our sole candidates for local extrema.

Next, use Second-Order Conditions via the Hessian matrix to classify each point:

$$H = \begin{pmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy} \end{pmatrix}$$

Compute the determinant $D = f_{xx} f_{yy} - (f_{xy})^2$ to reveal the local curvature.

Common pitfall: Never check $f_{xx} < 0$ and $f_{yy} < 0$ separately. A large cross-term $f_{xy}$ can still yield a saddle point. Only the determinant test captures full surface interaction.

The Hessian Verdicts

The determinant $D$ and $f_{xx}$ classify every critical point:

Economic application: A firm choosing output $q$ and advertising $a$ to maximise profit $\pi(q,a)$ relies on FOCs for candidates and the Hessian to confirm a local maximum. If the Hessian yields a saddle point, the firm is at a pass where altering mix improves profit.