Engineering Calculus Notes 376

# Engineering Calculus Notes 376 - 3.8.5 and then Lemma 3.8.4...

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364 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Theorem 3.8.6 (Second Derivative Test, Two Variables) . If f : R 2 R is C 2 and has a critical point at −→ x = −→ a , consider the determinant of the Hessian matrix 18 Δ = Δ 2 ( −→ a ) = f xx ( −→ a ) f yy ( −→ a ) f xy ( −→ a ) 2 , and its upper left entry Δ 1 ( −→ a ) = f xx . Then: 1. if Δ > 0 , then −→ a is a local extremum of f : (a) it is a local minimum if Δ 1 ( −→ a ) = f xx > 0 (b) it is a local maximum if Δ 1 ( −→ a ) = f xx < 0 ; 2. if Δ < 0 , then −→ a is not a local extremum of f ; 3. Δ = 0 does not give enough information to distinguish the possibilities. Proof. 1. We know that d 2 −→ a f is positive ( resp . negative) defnite by Proposition 3.8.5 , and then apply Proposition 3.8.3 . 2. Apply Proposition
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Unformatted text preview: 3.8.5 and then Lemma 3.8.4 in the same way. 3. Consider the Following three Functions: f ( x,y ) = ( x + y ) 2 = x 2 + 2 xy + y 2 g ( x,y ) = f ( x,y ) + y 4 = x 2 + 2 xy + y 2 + y 4 h ( x,y ) = f ( x,y ) − y 4 = x 2 + 2 xy + y 2 − y 4 . They all have second order contact at the origin, which is a critical point, and all have Hessian matrix A = b 1 1 1 1 B so all have Δ = 0. However: • f has a weak local minimum at the origin: the Function is non-negative everywhere, but equals zero along the whole line y = − x ; 18 sometimes called the discriminant of f...
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