Two Variable Extreme Values

Types

min, max, and saddle

Long Way

$$\begin{array}{rl}& \text{where }{P}_0=(x_0,y_0)\text{ is a critical point}\\ & \\ & a=f_{xx}^{\prime \prime }(x_0,y_0)\\ & b=f_{yy}^{\prime \prime }(x_0,y_0)\\ & c=f_{xy}^{\prime \prime }(x_0,y_0)\\ & \\ & {\lambda }_1,{\lambda }_2\text{ roots of (will be real):}\\ & \det \left(\begin{array}{cc}a-\lambda & c\\ c& b-\lambda \end{array}\right)=(a-\lambda )(b-\lambda )-c^2=0\\ & \\ & {\lambda }_1,{\lambda }_2>0⟹\text{min}\\ & {\lambda }_1,{\lambda }_2<0⟹\text{max}\\ & {\lambda }_1{\lambda }_2<0\text{ (diff signs)}⟹\text{saddle}\\ & {\lambda }_1=0\text{ or }{\lambda }_2=0⟹\text{no info}\end{array}$$

Shortcut

• $D=ab-c^2$ (Discriminant)
• $D=0$ → No Info
• $D<0$ → Saddle
• $D>0,a>0$ → Min
• $D>0,a>0$ → Max

With Taylor Polynomials

• Can find if the origin is an extrema
• Replace the elementary function with its power expansion
  • Use $u$ = some combination of $x$ and $y$
  • After the first term use Big O
• Plug it into $f(x,y)-f(0,0)$
• If what is outside of the |Big O in the neighborhood of (0,0)…
  • Always positive → Min
  • Always negative → Max
  • Both → Not an extrema