PX153 - G5 - critical points of a function of two variables

for a function of one variable

• expanding $f(x)$ around this point using a taylor series:

- $(x-x_c{)}^2$ is always positive
- for local maxima, $f(x)<f(x_c)⟹f^{″}<0$
- for local minima, $f(x)>f(x_c)⟹f^{″}>0$
- if $f^{″}=0$, might be point of inflection

for a function of two variables

• there are three types of critical points:
  • local maxima
  • local minima
  • saddle point: approached along some directions, they look like maxima, and along other directions, they look like minima
• at all critical points, both $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$
• for all maxima, all nearby points are lower in value; for all minima, all higher; and for saddle, some higher, some lower
• apply taylor series to second order, with the first order derivatives equal to zero
• we have, $(x-x_c=\mathrm{\Delta }x$ , $y-y_c=\mathrm{\Delta }y)$ :

- whose $f_{xx}...,etc$ are evaluated at $(x_c,y_c)$

• rearrange to be similar to the expression for a function of one variable, ie: square on $RHS$:

• completing the square:

• $(f_{xx}\mathrm{\Delta }x+f_{xy}\mathrm{\Delta }y{)}^2>0$ and $\mathrm{\Delta }{y}^2>0$ : always

• taking $\mathrm{\Delta }=f_{xx}{f}_{yy}-f_{xy}^2$ :

  • if $\mathrm{\Delta }>0$ and $\mathrm{\Delta }t>0$ :
    • if $f_{xx}>0⟹f(x,y)-f(x_c,y_c)>0$, then $(x_c,y_c)$ is a local minimum
    • if $f_{xx}<0⟹f(x,y)-f(x_c,y_c)<0$, then $(x_c,y_c)$ is a local maximum
  • if $\mathrm{\Delta }<0$, then the sign of $f(x,y)-f(x_c,y_c)$ can change depending on the direction, ie: saddle point
  • if $\mathrm{\Delta }=f_{xx}{f}_{yy}-f_{xy}^2=0$, then the test is inconclusive

• eg:

• eg:

• eg:

• eg: