14.7 Videos Guide

14.7a

• Definition of local maximum and local minimum values

  • If $f(a,\ b)\geq f(x,\ y)$ for all $(x,\ y)$ in an open disk, then $f(a,\ b)$ is a local maximum

  • If $f(a,\ b)\leq f(x,\ y)$ for all $(x,\ y)$ in an open disk, then $f(a,\ b)$ is a local minimum

• Definition of a critical point

  • A critical point is a point $(a,\ b)$ in the domain of $f$ such that
    $f_{x}(a,\ b)=f_{y}(a,\ b)=0$ or one of the first partial derivatives does not exist

• Second Derivatives Test and description of a saddle point

  • Let $(a,\ b)$ be a critical point of $f$ and let
    $D(a,\ b)=f_{xx}(a,\ b)f_{yy}(a,\ b)-\left\lbrack f_{xy}(a,\ b)\right\rbrack ^{2}$

    • If $D<0$, then $(a,\ b)$ is a saddle point

    • If $D>0$, then

      • if $f_{xx}(a,\ b)>0$, then $f(a,\ b)$ is a local minimum

      • if $f_{xx}(a,\ b)<0$, then $f(a,\ b)$ is a local maximum

    • If $D=0$, then the Second Derivatives Test is inconclusive

14.7b

Exercise:

• Find the local maximum and minimum values and saddle point(s) of the function. Then graph the surface using a window that shows these characteristics.
  $$f(x,\ y)=xye^{-\frac{\left(x^{2}+y^{2}\right)}{2}}$$

14.7c

• Absolute extrema

  • If $f(a,\ b)\geq f(x,\ y)$ for all $(x,\ y)$ in the domain of $f$, then $f(a,\ b)$ is the absolute maximum

  • If $f(a,\ b)\leq f(x,\ y)$ for all $(x,\ y)$ in the domain of $f$, then $f(a,\ b)$ is the absolute minimum

Exercise:

• Find the absolute maximum and minimum values of $f$ on the set $D$.
  $f(x,\ y)=x+y-xy$; $D$ is the closed triangular region with vertices $(0,\ 0)$, $(0,\ 2)$, and $(4,\ 0)$.