M310 L05 14'7 Maxima, Minima, Saddles

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M310 L05 14.7 Maxima, Minima, Saddles

• Polynomial Approximations
• Functionals
• Critical Points grad F 0
• Local Extremes
• Second Derivatives Test
• Absolute Extremes

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14.7 Polynomial Approximations
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14.7 Local Extremes
A functional has a local maximum at (a,b) iff
there is a neighborhood of (a,b) in which f(x,y)
is not more than f(a,b). A functional has a local
minimum at (a,b) iff there is a neighborhood of
(a,b) in which f(x,y) is not less than
f(a,b). THEOREM If f has either a local maximum
or a local minimum at (a,b), then grad f(a,b)
0. Remark grad f(a,b) 0 does not assure
that f(a,b) is an extreme value of
f. Proof To prove fx(a,b)0, consider
f(x,b). To prove fy(a,b)0, consider f(a,y).
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14.7 1 Critical Points of a poly in 2 var
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14.7 2 Critical Points of a poly in 2 var
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14.7 Second Derivatives Test
The determinant H is called the Hessian of f .
It is understood that the partial derivatives
are all continuous at some critical point (a,b)
and all evaluated there. THEOREM If the
second partial derivatives of f are all
continuous at the critical point (a,b), then (the
cross-partials are equal and) (a) If Hgt0 and
fxxgt0, then f has a minimum at (a,b). (b) If
Hgt0 and fxxlt0, then f has a maximum at
(a,b). (c) If Hlt0, then f has NO local
extreme, but has a saddle. REMARK If Hgt0, then
fxxfyygt0.
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14.7 Proof of Second Derivatives Test
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14.7 3 Find Local Extremes, Saddles
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14.7 4 Find and Classify Critical Pts