Section 15'1 Local Extrema

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Section 15.1Local Extrema
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Definition of Local Extrema

• Let P be a point in the domain of f. Then
• f has a local maximum at the point P0 if f(P0 )
  f(P) for all points P near P0
• f has a local minimum at the point P0 if f(P0 )
  f(P) for all points P near P0
• These definitions are similar to those we saw
  back in single variable calculus
• How did we determine local extrema back then?

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Definition of Critical Points

• Suppose a function has a local maximum at P0
  which does not lie on the boundary of the domain
• Recall that the gradient vector (if it is
  defined and nonzero) points in the direction of
  maximum increase
• What is direction of the maximum rate of increase
  at P0?
• Thus we have critical points where the gradient
  is the zero vector or is undefined

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Definition of Critical Points

• P0 is a critical point of f if

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Example

• Find the critical points of the following
  functions
• We will use maple to determine what is going on
  at these critical points
• We need a test for determining whats going on at
  our critical points

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Second Derivative Test

• Old Stuff Suppose f has a critical point at x
  a, continuous first and second derivatives (at x
  a) and f(a) 0, then
• Using a 2nd degree Taylor polynomial we have
• Which is a quadratic whose concavity depends on
  the sign of f(a)

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Second Derivative Test for y f(x)

• Suppose f has a critical point at x a,
  continuous first and second derivatives (at x
  a) and f(a) 0, then
• f(a) gt 0 ? f(a) is a local minimum
• f(a) lt 0 ? f(a) is a local maximum
• f(a) 0 ? no conclusion can be drawn

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Second Derivative Test for Functions of Two
Variables

• Suppose Q is defined by
• Then
• Note

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• Based on this form, what do you think we need in
  order to have our graph be concave up in the x
  direction? y direction?

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• Given
• If a gt 0 and 4ac-b2 gt 0 ? Q has a local min at
  (0,0)
• If a lt 0 and 4ac-b2 gt 0 ? Q has a local max at
  (0,0)
• If a gt 0 and 4ac-b2 lt 0 ? Q has a saddle point at
  (0,0)
• If a lt 0 and 4ac-b2 lt 0 ? Q has a saddle point at
  (0,0)
• If 4ac-b2 0 no conclusion can be drawn

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Second Derivative Test for Functions of Two
Variables

• Suppose f(x,y) has continuous 1st and 2nd parital
  derivatives at (0,0), that (0,0) is a critical
  point and f(0,0) 0. From 14.7 we know

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• Given
• we have
• So
• Thus we have our discriminant and we can restate
  our previous conclusions

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Second Derivative Test for Functions of Two
Variables

• If f has a critical point at P0 then the
  discriminant is
• If D gt 0 and
• fxx(P0) gt 0 ? f(P0) is a local minimum
• fxx(P0) lt 0 ? f(P0) is a local maximum
• If D lt 0 ? f(P0) is a saddle point
• If D 0 ? no conclusion can be drawn from the
  second derivative test
• Note Since we solved for this test by completing
  the square in terms of x, we use the second
  partial of x

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Examples

• 2 from the book
• 4 from the book