Local Extreme Values

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• Extreme Values
  • Section 14.7
    • Local minimum, maximum
    • Saddle point: in every disk around P₀ some points have f(x,y) < f(P₀) and some have f(x,y) > f(P₀)
    • Critical points: f_x = f_y = 0, or derivatives don’t exist
      • Extrema can exist here
    • 2nd derivative test for local extrema
    • To find absolute minima, maxima on closed bounded regions
      • List interior points that might be extrema
      • List boundary points that might be extrema
      • Find most extreme items in both lists
  • Examples
    • z = x² + y²
      
    • z = x³/3 - x² + y² + 2y
      
      • Plot this to see what it looks like in detail
        
    • z = x³y³
      
      • This function has infinitely many critical points, and they’re all saddle points, though you have to use the definition of saddle point to realize that
        
  • Problem set
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  • Absolute maxima and minima
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