Comparing max/min rules for functions y = f (x) and z = f (x, y)
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f has an absolute max at a in D if f (x) ≤ f (a)
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for all x in D. (Absolute min similarly.)
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f has a relative max at a in D if there is an h > 0 k
such that for all |x − a| < h, x is in D
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and f (x) ≤ f (a). (Relative min similarly.)
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f has a critical point at a if
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either f 0 (a) = 0 or f 0 (a) DNE.
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If f has a relative max or min at a,
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then f has a critical point at a.
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NOT every critical point is a relative max/min.
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Example: y = x3 at the origin.
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Second Derivative Test: If f 0 (a) = 0 and if
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f 00 is continuous at a,
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then the Second Derivative Test can (sometimes)
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classify this critical point as a relative max
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or relative min.
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If D is a closed interval, then f will have
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an absolute max and an absolute min on D.
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Furthermore, each of these either occurs at an
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endpoint of D or is a local max/min
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(and thus a critical point).
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First Derivative Test. Let f 0 (a) = 0. If f 0 changes
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sign from − to + as x passes a, then f has a
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local min at a. Similarly, if f 0 changes
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sign from + to − as x passes a, then f has a
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local max at a.
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y = f (x), continuous on domain D of numbers:
z = f (x, y), continuous on domain D of points in the plane:
f has an absolute max at (a, b) in D if f (x, y) ≤ f (a, b)
for all (x, y) in D. (Absolute min similarly.)
f has a relative max at (a, b) in D if there is an h > 0
such that for all k(x, y) − (a, b)k < h, (x, y) is in D
and f (x, y) ≤ f (a, b). (Relative min similarly.)
f has a critical point at (a, b) if
either fx (a, b) = fy (a, b) = 0 or at least one of these
partial derivatives DNE.
If f has a relative max or min at (a, b),
then f has a critical point at (a, b).
NOT every critical point is a relative max/min.
Example: z = x2 − y 2 at the origin. (Saddle point.)
Second Derivative Test: If fx (a, b) = fy (a, b) = 0 and if
fxx , fxy , fyx , fyy are all continuous at (a, b),
then the Second Derivative Test can (sometimes)
classify this critical point as a relative max,
relative min, or saddle point.
If D is closed and bounded, then f will have
an absolute max and an absolute min on D.
Furthermore, each of these either occurs at a
boundary point of D or is a local max/min
(and thus a critical point).
There is no First Derivative Test.