Engineering Calculus Notes 401

# Engineering Calculus Notes 401 - no information As an...

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3.9. THE PRINCIPAL AXIS THEOREM 389 Theorem 3.9.8 (Second Derivative Test, Three Variables) . Suppose the C 2 function f ( −→ x ) = f ( x,y,z ) has a critical point at −→ x = −→ a . Consider the following three quantities: Δ 1 = f xx ( −→ a ) Δ 2 = f xx ( −→ a ) f yy ( −→ a ) f xy ( −→ a ) 2 Δ 3 = det Hf ( −→ a ) . 1. If Δ k > 0 for k = 1 , 2 , 3 , then f ( −→ x ) has a local minimum at −→ x = −→ a . 2. If ( 1) k Δ k > 0 for k = 1 , 2 , 3 (i.e., , Δ 2 > 0 while Δ 1 < 0 and Δ 3 < 0 ), then f ( −→ x ) has a local maximum at −→ x = −→ a . 3. If all three quantities are nonzero but neither of the preceding conditions holds, then f ( −→ x ) does not have a local extremum at −→ x = −→ a . A word of warning: when one of these quantities equals zero, this test gives
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Unformatted text preview: no information. As an example of the use of Theorem 3.9.8 , consider the function f ( x,y,z ) = 2 x 2 + 2 y 2 + 2 z 2 + 2 xy + 2 xz + 2 yz − 6 x + 2 y + 4 z ; its partial derivatives are f x ( x,y,z ) = 4 x + 2 y + 2 z − 6 f y ( x,y,z ) = 4 y + 2 x + 2 z + 2 f z ( x,y,z ) = 4 z + 2 y + 2 x + 4 . Setting all of these equal to zero, we have the system of equations 4 x +2 y +2 z = 6 2 x +4 y +2 z = − 2 2 x +2 y +4 z = − 4 whose only solution is x = 3 y = − 1 z = − 2 ....
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