Engineering Calculus Notes 372

Engineering Calculus Notes 372 - K> 0 be the constant...

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360 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Using Lemma 3.8.2 and Taylor’s theorem (Proposition 3.7.2 ), we can show that a critical point with definite Hessian is a local extremum. Proposition 3.8.3. Suppose f is a C 2 function and −→ a is a critical point for f where the Hessian form d 2 −→ a f is definite. Then f has a local extremum at −→ a : If d 2 −→ a f is positive definite, then f has a local minimum at −→ a ; If d 2 −→ a f is negative definite, then f has a local maximum at −→ a . Proof. The fact that the quadratic approximation T 2 −→ a f ( −→ x ) has second order contact with f ( −→ x ) at −→ x = −→ a can be written in the form f ( −→ x ) = T 2 −→ a f ( −→ x ) + ε ( −→ x ) bardbl −→ x −→ a bardbl 2 , where lim −→ x −→ a ε ( −→ x ) = 0 . Since −→ a is a critical point, d −→ a f ( −→ x ) = 0, so T 2 −→ a f ( −→ x ) = f ( −→ a ) + 1 2 d 2 −→ a f ( −→ x ) , or f ( −→ x ) f ( −→ a ) = 1 2 d 2 −→ a f ( −→ x ) + ε ( −→ x ) bardbl△ −→ x bardbl 2 . Suppose d 2 −→ a f is positive definite, and let K> 0 be the constant given in
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Unformatted text preview: K > 0 be the constant given in Lemma 3.8.2 , such that d 2 −→ a f ( △ −→ x ) ≥ K b△ −→ x b 2 . Since ε ( −→ x ) → 0 as −→ x → −→ a , For b△ −→ x b su±ciently small, we have | ε ( −→ x ) | < K 4 and hence f ( −→ x ) − f ( −→ a ) ≥ { K 2 − ε ( −→ x ) }b△ −→ x b 2 > K 4 b△ −→ x b 2 > or f ( −→ x ) > f ( −→ a ) For −→ x n = −→ a ( b△ −→ x b su±ciently small). The argument when d 2 −→ a f is negative defnite is analogous (Exercise 4a ). An analogous argument (Exercise 4b ) gives Lemma 3.8.4. If d 2 −→ a f takes both positive and negative values at the critical point −→ x = −→ a of f , then f does not have a local extremum at −→ x = −→ a ....
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