Business I / Unconstrained Multivariable Extrema

Practice question · Put in order

Order the steps of finding and classifying a multivariable extremum.

Set $f_x = 0$ and $f_y = 0$ ; solve for critical points
Compute $f_{xx}$ , $f_{yy}$ , $f_{xy}$ at each critical point
Compute $f_x$ and $f_y$
Classify: sign of $D$ and $f_{xx}$ determine max/min/saddle
Compute $D = f_{xx} f_{yy} - f_{xy}^2$

Hints

Critical points must be found before they can be classified.
The second derivatives are only needed after the candidates exist.

Show the answer

Compute $f_x$ and $f_y$
Set $f_x = 0$ and $f_y = 0$ ; solve for critical points
Compute $f_{xx}$ , $f_{yy}$ , $f_{xy}$ at each critical point
Compute $D = f_{xx} f_{yy} - f_{xy}^2$
Classify: sign of $D$ and $f_{xx}$ determine max/min/saddle

Why

Differentiate, solve, differentiate again, compute the determinant, classify, the complete recipe.

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