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Unformatted text preview: 5.2 Properties of the derivative 73 5.2 Properties of the derivative 5.2.1 Local properties Definition 5.2.1. f is monotone increasing at x iff f ( s ) ≤ f ( x ) ≤ f ( t ) for a < s < x < t < b . f is strictly increasing at x iff f ( s ) < f ( x ) < f ( t ) for a < s < x < t < b . Proposition 5.2.2. f is (monotone or strictly) increasing on ( a,b ) iff f is orderpreserving on ( a,b ) . Proof. Immediate. Definition 5.2.3. f has a local maximum at x iff f ( t ) ≤ f ( x ), for all t ∈ ( x ε,x + ε ). f has a strict local maximum at x iff f ( t ) > f ( x ), for all t 6 = x in ( x ε,x + ε ). Theorem 5.2.4. If x is a local max or min of f , then f ( x ) = 0 . Proof. Choose δ such that a < x δ < x < x + δ < b . Then for x δ < t < x , we have f ( t ) f ( x ) t x ≥ . Letting t → x , get f ( x ) ≥ 0. Similarly for the other inequality. Example 5.2.1. f ( x ) = x 3 is strictly increasing, but has derivative 0 at 0....
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 Fall '11
 Wong
 Derivative, Probability theory, Order theory, Monotonic function, Convex function, 413 g

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