Using derivatives to locate maxima and minima of a surface

For a function $f:R^2\to R$,

• $f(a,b)$ is the global maximum of $f$ if $f(x,y)\le f(a,b)\mathrm{\forall }(x,y)\in R^2$
• $f(a,b)$ is the global minimum of $f$ if $f(x,y)\ge f(a,b)\mathrm{\forall }(x,y)\in R^2$

Finding possible global extrema

Start with the interior of $D$

1. Find all critical points of $f$ existing on the interior of $D$
2. Evaluate $f$ at those points
   Work on the boundary of $D$
3. Evaluate $f$ along the boundaries of $D$ to determine if there are any extrema there
4. Evaluate $f$ along any endpoints formed by the boundary of $D$

• if there are no critical points ($\frac{d}{dx}\ne 0$ everywhere), then endpoints will be max/min of the line
  Conclusion

5. The global maximum is the biggest value found in 2-4, global minimum is the smallest value found in 2-4

Lagrange Multipliers

Lagrangian function

Critical Points of the Lagrangian are found by solving $\mathrm{\nabla }L=\tilde{0}$

• the scalar $\lambda$ is called the lagrangian multiplier (?)

Lagrangian can be extended to multiple constraints